Polycarpus likes giving presents to Paraskevi. He has bought two chocolate bars, each of them has the shape of a segmented rectangle. The first bar is a1 × b1 segments large and the second one is a2 × b2 segments large.Polycarpus wants to give Paraskevi one of the bars at the lunch break and eat the other one himself. Besides, he wants to show that Polycarpus's mind and Paraskevi's beauty are equally matched, so the two bars must have the same number of squares.To make the bars have the same number of squares, Polycarpus eats a little piece of chocolate each minute. Each minute he does the following:  he either breaks one bar exactly in half (vertically or horizontally) and eats exactly a half of the bar,  or he chips of exactly one third of a bar (vertically or horizontally) and eats exactly a third of the bar. In the first case he is left with a half, of the bar and in the second case he is left with two thirds of the bar.Both variants aren't always possible, and sometimes Polycarpus cannot chip off a half nor a third. For example, if the bar is 16 × 23, then Polycarpus can chip off a half, but not a third. If the bar is 20 × 18, then Polycarpus can chip off both a half and a third. If the bar is 5 × 7, then Polycarpus cannot chip off a half nor a third.What is the minimum number of minutes Polycarpus needs to make two bars consist of the same number of squares? Find not only the required minimum number of minutes, but also the possible sizes of the bars after the process.

The first line of the input contains integers a1, b1 (1 ≤ a1, b1 ≤ 109) — the initial sizes of the first chocolate bar. The second line of the input contains integers a2, b2 (1 ≤ a2, b2 ≤ 109) — the initial sizes of the second bar. You can use the data of type int64 (in Pascal), long long (in С++), long (in Java) to process large integers (exceeding 231 - 1).

In the first line print m — the sought minimum number of minutes. In the second and third line print the possible sizes of the bars after they are leveled in m minutes. Print the sizes using the format identical to the input format. Print the sizes (the numbers in the printed pairs) in any order. The second line must correspond to the first bar and the third line must correspond to the second bar. If there are multiple solutions, print any of them. If there is no solution, print a single line with integer -1.


Let's write all the positive integer numbers one after another from $$$1$$$ without any delimiters (i.e. as a single string). It will be the infinite sequence starting with 123456789101112131415161718192021222324252627282930313233343536...Your task is to print the $$$k$$$-th digit of this sequence.

The first and only line contains integer $$$k$$$ ($$$1 \le k \le 10000$$$) — the position to process ($$$1$$$-based index).

Print the $$$k$$$-th digit of the resulting infinite sequence.


As a tradition, every year before IOI all the members of Natalia Fan Club are invited to Malek Dance Club to have a fun night together. Malek Dance Club has 2n members and coincidentally Natalia Fan Club also has 2n members. Each member of MDC is assigned a unique id i from 0 to 2n - 1. The same holds for each member of NFC.One of the parts of this tradition is one by one dance, where each member of MDC dances with a member of NFC. A dance pair is a pair of numbers (a, b) such that member a from MDC dances with member b from NFC.The complexity of a pairs' assignment is the number of pairs of dancing pairs (a, b) and (c, d) such that a < c and b > d.You are given a binary number of length n named x. We know that member i from MDC dances with member  from NFC. Your task is to calculate the complexity of this assignment modulo 1000000007 (109 + 7).Expression  denotes applying «XOR» to numbers x and y. This operation exists in all modern programming languages, for example, in C++ and Java it denotes as «^», in Pascal — «xor».

The first line of input contains a binary number x of lenght n, (1 ≤ n ≤ 100). This number may contain leading zeros.

Print the complexity of the given dance assignent modulo 1000000007 (109 + 7).


You are given a rectangular cake, represented as an r × c grid. Each cell either has an evil strawberry, or is empty. For example, a 3 × 4 cake may look as follows:  The cakeminator is going to eat the cake! Each time he eats, he chooses a row or a column that does not contain any evil strawberries and contains at least one cake cell that has not been eaten before, and eats all the cake cells there. He may decide to eat any number of times.Please output the maximum number of cake cells that the cakeminator can eat.

The first line contains two integers r and c (2 ≤ r, c ≤ 10), denoting the number of rows and the number of columns of the cake. The next r lines each contains c characters — the j-th character of the i-th line denotes the content of the cell at row i and column j, and is either one of these:    '.' character denotes a cake cell with no evil strawberry;  'S' character denotes a cake cell with an evil strawberry.

Output the maximum number of cake cells that the cakeminator can eat.

NoteFor the first example, one possible way to eat the maximum number of cake cells is as follows (perform 3 eats).


This is an easier version of the problem. In this version, $$$n \le 500$$$.Vasya is an experienced developer of programming competitions' problems. As all great minds at some time, Vasya faced a creative crisis. To improve the situation, Petya gifted him a string consisting of opening and closing brackets only. Petya believes, that the beauty of the bracket string is a number of its cyclical shifts, which form a correct bracket sequence.To digress from his problems, Vasya decided to select two positions of the string (not necessarily distinct) and swap characters located at this positions with each other. Vasya will apply this operation exactly once. He is curious what is the maximum possible beauty he can achieve this way. Please help him.We remind that bracket sequence $$$s$$$ is called correct if:   $$$s$$$ is empty;  $$$s$$$ is equal to "($$$t$$$)", where $$$t$$$ is correct bracket sequence;  $$$s$$$ is equal to $$$t_1 t_2$$$, i.e. concatenation of $$$t_1$$$ and $$$t_2$$$, where $$$t_1$$$ and $$$t_2$$$ are correct bracket sequences. For example, "(()())", "()" are correct, while ")(" and "())" are not.The cyclical shift of the string $$$s$$$ of length $$$n$$$ by $$$k$$$ ($$$0 \leq k < n$$$) is a string formed by a concatenation of the last $$$k$$$ symbols of the string $$$s$$$ with the first $$$n - k$$$ symbols of string $$$s$$$. For example, the cyclical shift of string "(())()" by $$$2$$$ equals "()(())".Cyclical shifts $$$i$$$ and $$$j$$$ are considered different, if $$$i \ne j$$$.

The first line contains an integer $$$n$$$ ($$$1 \le n \le 500$$$), the length of the string. The second line contains a string, consisting of exactly $$$n$$$ characters, where each of the characters is either "(" or ")".

The first line should contain a single integer — the largest beauty of the string, which can be achieved by swapping some two characters. The second line should contain integers $$$l$$$ and $$$r$$$ ($$$1 \leq l, r \leq n$$$) — the indices of two characters, which should be swapped in order to maximize the string's beauty. In case there are several possible swaps, print any of them.

NoteIn the first example, we can swap $$$7$$$-th and $$$8$$$-th character, obtaining a string "()()()()()". The cyclical shifts by $$$0, 2, 4, 6, 8$$$ of this string form a correct bracket sequence.In the second example, after swapping $$$5$$$-th and $$$10$$$-th character, we obtain a string ")(())()()(()". The cyclical shifts by $$$11, 7, 5, 3$$$ of this string form a correct bracket sequence.In the third example, swap of any two brackets results in $$$0$$$ cyclical shifts being correct bracket sequences.


Tokitsukaze is playing a game derivated from Japanese mahjong. In this game, she has three tiles in her hand. Each tile she owns is a suited tile, which means it has a suit (manzu, pinzu or souzu) and a number (a digit ranged from $$$1$$$ to $$$9$$$). In this problem, we use one digit and one lowercase letter, which is the first character of the suit, to represent a suited tile. All possible suited tiles are represented as 1m, 2m, $$$\ldots$$$, 9m, 1p, 2p, $$$\ldots$$$, 9p, 1s, 2s, $$$\ldots$$$, 9s.In order to win the game, she must have at least one mentsu (described below) in her hand, so sometimes she should draw extra suited tiles. After drawing a tile, the number of her tiles increases by one. She can draw any tiles she wants, including those already in her hand.Do you know the minimum number of extra suited tiles she needs to draw so that she can win?Here are some useful definitions in this game:  A mentsu, also known as meld, is formed by a koutsu or a shuntsu;  A koutsu, also known as triplet, is made of three identical tiles, such as [1m, 1m, 1m], however, [1m, 1p, 1s] or [1m, 4m, 7m] is NOT a koutsu;  A shuntsu, also known as sequence, is made of three sequential numbered tiles in the same suit, such as [1m, 2m, 3m] and [5s, 7s, 6s], however, [9m, 1m, 2m] or [1m, 2p, 3s] is NOT a shuntsu. Some examples:   [2m, 3p, 2s, 4m, 1s, 2s, 4s] — it contains no koutsu or shuntsu, so it includes no mentsu;  [4s, 3m, 3p, 4s, 5p, 4s, 5p] — it contains a koutsu, [4s, 4s, 4s], but no shuntsu, so it includes a mentsu;  [5p, 5s, 9m, 4p, 1s, 7p, 7m, 6p] — it contains no koutsu but a shuntsu, [5p, 4p, 6p] or [5p, 7p, 6p], so it includes a mentsu. Note that the order of tiles is unnecessary and you can assume the number of each type of suited tiles she can draw is infinite.

The only line contains three strings — the tiles in Tokitsukaze's hand. For each string, the first character is a digit ranged from $$$1$$$ to $$$9$$$ and the second character is m, p or s.

Print a single integer — the minimum number of extra suited tiles she needs to draw.

NoteIn the first example, Tokitsukaze already has a shuntsu.In the second example, Tokitsukaze already has a koutsu.In the third example, Tokitsukaze can get a shuntsu by drawing one suited tile — 1p or 4p. The resulting tiles will be [3p, 9m, 2p, 1p] or [3p, 9m, 2p, 4p].


Mike has a frog and a flower. His frog is named Xaniar and his flower is named Abol. Initially(at time 0), height of Xaniar is h1 and height of Abol is h2. Each second, Mike waters Abol and Xaniar.  So, if height of Xaniar is h1 and height of Abol is h2, after one second height of Xaniar will become  and height of Abol will become  where x1, y1, x2 and y2 are some integer numbers and  denotes the remainder of a modulo b.Mike is a competitive programmer fan. He wants to know the minimum time it takes until height of Xania is a1 and height of Abol is a2.Mike has asked you for your help. Calculate the minimum time or say it will never happen.

The first line of input contains integer m (2 ≤ m ≤ 106). The second line of input contains integers h1 and a1 (0 ≤ h1, a1 < m). The third line of input contains integers x1 and y1 (0 ≤ x1, y1 < m). The fourth line of input contains integers h2 and a2 (0 ≤ h2, a2 < m). The fifth line of input contains integers x2 and y2 (0 ≤ x2, y2 < m). It is guaranteed that h1 ≠ a1 and h2 ≠ a2.

Print the minimum number of seconds until Xaniar reaches height a1 and Abol reaches height a2 or print -1 otherwise.

NoteIn the first sample, heights sequences are following:Xaniar: Abol:


Imp is in a magic forest, where xorangles grow (wut?)  A xorangle of order n is such a non-degenerate triangle, that lengths of its sides are integers not exceeding n, and the xor-sum of the lengths is equal to zero. Imp has to count the number of distinct xorangles of order n to get out of the forest. Formally, for a given integer n you have to find the number of such triples (a, b, c), that:  1 ≤ a ≤ b ≤ c ≤ n;  , where  denotes the bitwise xor of integers x and y.  (a, b, c) form a non-degenerate (with strictly positive area) triangle.

The only line contains a single integer n (1 ≤ n ≤ 2500).

Print the number of xorangles of order n.

NoteThe only xorangle in the first sample is (3, 5, 6).


n participants of the competition were split into m teams in some manner so that each team has at least one participant. After the competition each pair of participants from the same team became friends.Your task is to write a program that will find the minimum and the maximum number of pairs of friends that could have formed by the end of the competition.

The only line of input contains two integers n and m, separated by a single space (1 ≤ m ≤ n ≤ 109) — the number of participants and the number of teams respectively.

The only line of the output should contain two integers kmin and kmax — the minimum possible number of pairs of friends and the maximum possible number of pairs of friends respectively.

NoteIn the first sample all the participants get into one team, so there will be exactly ten pairs of friends.In the second sample at any possible arrangement one team will always have two participants and the other team will always have one participant. Thus, the number of pairs of friends will always be equal to one.In the third sample minimum number of newly formed friendships can be achieved if participants were split on teams consisting of 2 people, maximum number can be achieved if participants were split on teams of 1, 1 and 4 people.


Ayoub had an array $$$a$$$ of integers of size $$$n$$$ and this array had two interesting properties:   All the integers in the array were between $$$l$$$ and $$$r$$$ (inclusive).  The sum of all the elements was divisible by $$$3$$$. Unfortunately, Ayoub has lost his array, but he remembers the size of the array $$$n$$$ and the numbers $$$l$$$ and $$$r$$$, so he asked you to find the number of ways to restore the array. Since the answer could be very large, print it modulo $$$10^9 + 7$$$ (i.e. the remainder when dividing by $$$10^9 + 7$$$). In case there are no satisfying arrays (Ayoub has a wrong memory), print $$$0$$$.

The first and only line contains three integers $$$n$$$, $$$l$$$ and $$$r$$$ ($$$1 \le n \le 2 \cdot 10^5 , 1 \le l \le r \le 10^9$$$) — the size of the lost array and the range of numbers in the array.

Print the remainder when dividing by $$$10^9 + 7$$$ the number of ways to restore the array.

NoteIn the first example, the possible arrays are : $$$[1,2], [2,1], [3, 3]$$$.In the second example, the only possible array is $$$[2, 2, 2]$$$.


Hideo Kojima has just quit his job at Konami. Now he is going to find a new place to work. Despite being such a well-known person, he still needs a CV to apply for a job.During all his career Hideo has produced n games. Some of them were successful, some were not. Hideo wants to remove several of them (possibly zero) from his CV to make a better impression on employers. As a result there should be no unsuccessful game which comes right after successful one in his CV.More formally, you are given an array s1, s2, ..., sn of zeros and ones. Zero corresponds to an unsuccessful game, one — to a successful one. Games are given in order they were produced, and Hideo can't swap these values. He should remove some elements from this array in such a way that no zero comes right after one.Besides that, Hideo still wants to mention as much games in his CV as possible. Help this genius of a man determine the maximum number of games he can leave in his CV.

The first line contains one integer number n (1 ≤ n ≤ 100). The second line contains n space-separated integer numbers s1, s2, ..., sn (0 ≤ si ≤ 1). 0 corresponds to an unsuccessful game, 1 — to a successful one.

Print one integer — the maximum number of games Hideo can leave in his CV so that no unsuccessful game comes after a successful one.


The elections to Berland parliament are happening today. Voting is in full swing!Totally there are n candidates, they are numbered from 1 to n. Based on election results k (1 ≤ k ≤ n) top candidates will take seats in the parliament.After the end of the voting the number of votes for each candidate is calculated. In the resulting table the candidates are ordered by the number of votes. In case of tie (equal number of votes) they are ordered by the time of the last vote given. The candidate with ealier last vote stands higher in the resulting table.So in the resulting table candidates are sorted by the number of votes (more votes stand for the higher place) and if two candidates have equal number of votes they are sorted by the time of last vote (earlier last vote stands for the higher place).There is no way for a candidate with zero votes to take a seat in the parliament. So it is possible that less than k candidates will take a seat in the parliament.In Berland there are m citizens who can vote. Each of them will vote for some candidate. Each citizen will give a vote to exactly one of n candidates. There is no option "against everyone" on the elections. It is not accepted to spoil bulletins or not to go to elections. So each of m citizens will vote for exactly one of n candidates.At the moment a citizens have voted already (1 ≤ a ≤ m). This is an open election, so for each citizen it is known the candidate for which the citizen has voted. Formally, the j-th citizen voted for the candidate gj. The citizens who already voted are numbered in chronological order; i.e. the (j + 1)-th citizen voted after the j-th.The remaining m - a citizens will vote before the end of elections, each of them will vote for one of n candidates.Your task is to determine for each of n candidates one of the three possible outcomes:  a candidate will be elected to the parliament regardless of votes of the remaining m - a citizens;  a candidate has chance to be elected to the parliament after all n citizens have voted;  a candidate has no chances to be elected to the parliament regardless of votes of the remaining m - a citizens.

The first line contains four integers n, k, m and a (1 ≤ k ≤ n ≤ 100, 1 ≤ m ≤ 100, 1 ≤ a ≤ m) — the number of candidates, the number of seats in the parliament, the number of Berland citizens and the number of citizens who already have voted. The second line contains a sequence of a integers g1, g2, ..., ga (1 ≤ gj ≤ n), where gj is the candidate for which the j-th citizen has voted. Citizens who already voted are numbered in increasing order of voting times.

Print the sequence consisting of n integers r1, r2, ..., rn where:   ri = 1 means that the i-th candidate is guaranteed to take seat in the parliament regardless of votes of the remaining m - a citizens;  ri = 2 means that the i-th candidate has a chance to take a seat in the parliament, i.e. the remaining m - a citizens can vote in such a way that the candidate will take a seat in the parliament;  ri = 3 means that the i-th candidate will not take a seat in the parliament regardless of votes of the remaining m - a citizens.


Vova, the Ultimate Thule new shaman, wants to build a pipeline. As there are exactly n houses in Ultimate Thule, Vova wants the city to have exactly n pipes, each such pipe should be connected to the water supply. A pipe can be connected to the water supply if there's water flowing out of it. Initially Vova has only one pipe with flowing water. Besides, Vova has several splitters.A splitter is a construction that consists of one input (it can be connected to a water pipe) and x output pipes. When a splitter is connected to a water pipe, water flows from each output pipe. You can assume that the output pipes are ordinary pipes. For example, you can connect water supply to such pipe if there's water flowing out from it. At most one splitter can be connected to any water pipe.  The figure shows a 4-output splitter Vova has one splitter of each kind: with 2, 3, 4, ..., k outputs. Help Vova use the minimum number of splitters to build the required pipeline or otherwise state that it's impossible.Vova needs the pipeline to have exactly n pipes with flowing out water. Note that some of those pipes can be the output pipes of the splitters.

The first line contains two space-separated integers n and k (1 ≤ n ≤ 1018, 2 ≤ k ≤ 109). Please, do not use the %lld specifier to read or write 64-bit integers in С++. It is preferred to use the cin, cout streams or the %I64d specifier.

Print a single integer — the minimum number of splitters needed to build the pipeline. If it is impossible to build a pipeline with the given splitters, print -1.


Several ages ago Berland was a kingdom. The King of Berland adored math. That's why, when he first visited one of his many palaces, he first of all paid attention to the floor in one hall. The floor was tiled with hexagonal tiles.The hall also turned out hexagonal in its shape. The King walked along the perimeter of the hall and concluded that each of the six sides has a, b, c, a, b and c adjacent tiles, correspondingly.To better visualize the situation, look at the picture showing a similar hexagon for a = 2, b = 3 and c = 4.  According to the legend, as the King of Berland obtained the values a, b and c, he almost immediately calculated the total number of tiles on the hall floor. Can you do the same?

The first line contains three integers: a, b and c (2 ≤ a, b, c ≤ 1000).

Print a single number — the total number of tiles on the hall floor.


You have a fraction . You need to find the first occurrence of digit c into decimal notation of the fraction after decimal point.

The first contains three single positive integers a, b, c (1 ≤ a < b ≤ 105, 0 ≤ c ≤ 9).

Print position of the first occurrence of digit c into the fraction. Positions are numbered from 1 after decimal point. It there is no such position, print -1.

NoteThe fraction in the first example has the following decimal notation: . The first zero stands on second position.The fraction in the second example has the following decimal notation: . There is no digit 7 in decimal notation of the fraction.


There are three friend living on the straight line Ox in Lineland. The first friend lives at the point x1, the second friend lives at the point x2, and the third friend lives at the point x3. They plan to celebrate the New Year together, so they need to meet at one point. What is the minimum total distance they have to travel in order to meet at some point and celebrate the New Year?It's guaranteed that the optimal answer is always integer.

The first line of the input contains three distinct integers x1, x2 and x3 (1 ≤ x1, x2, x3 ≤ 100) — the coordinates of the houses of the first, the second and the third friends respectively.

Print one integer — the minimum total distance the friends need to travel in order to meet together.

NoteIn the first sample, friends should meet at the point 4. Thus, the first friend has to travel the distance of 3 (from the point 7 to the point 4), the second friend also has to travel the distance of 3 (from the point 1 to the point 4), while the third friend should not go anywhere because he lives at the point 4.


HQ9+ is a joke programming language which has only four one-character instructions: "H" prints "Hello, World!", "Q" prints the source code of the program itself, "9" prints the lyrics of "99 Bottles of Beer" song,  "+" increments the value stored in the internal accumulator.Instructions "H" and "Q" are case-sensitive and must be uppercase. The characters of the program which are not instructions are ignored.You are given a program written in HQ9+. You have to figure out whether executing this program will produce any output.

The input will consist of a single line p which will give a program in HQ9+. String p will contain between 1 and 100 characters, inclusive. ASCII-code of each character of p will be between 33 (exclamation mark) and 126 (tilde), inclusive.

Output "YES", if executing the program will produce any output, and "NO" otherwise.

NoteIn the first case the program contains only one instruction — "H", which prints "Hello, World!".In the second case none of the program characters are language instructions.


Alice and Bob have decided to play the game "Rock, Paper, Scissors". The game consists of several rounds, each round is independent of each other. In each round, both players show one of the following things at the same time: rock, paper or scissors. If both players showed the same things then the round outcome is a draw. Otherwise, the following rules applied:  if one player showed rock and the other one showed scissors, then the player who showed rock is considered the winner and the other one is considered the loser;  if one player showed scissors and the other one showed paper, then the player who showed scissors is considered the winner and the other one is considered the loser;  if one player showed paper and the other one showed rock, then the player who showed paper is considered the winner and the other one is considered the loser. Alice and Bob decided to play exactly $$$n$$$ rounds of the game described above. Alice decided to show rock $$$a_1$$$ times, show scissors $$$a_2$$$ times and show paper $$$a_3$$$ times. Bob decided to show rock $$$b_1$$$ times, show scissors $$$b_2$$$ times and show paper $$$b_3$$$ times. Though, both Alice and Bob did not choose the sequence in which they show things. It is guaranteed that $$$a_1 + a_2 + a_3 = n$$$ and $$$b_1 + b_2 + b_3 = n$$$.Your task is to find two numbers:  the minimum number of round Alice can win;  the maximum number of rounds Alice can win.

The first line of the input contains one integer $$$n$$$ ($$$1 \le n \le 10^{9}$$$) — the number of rounds. The second line of the input contains three integers $$$a_1, a_2, a_3$$$ ($$$0 \le a_i \le n$$$) — the number of times Alice will show rock, scissors and paper, respectively. It is guaranteed that $$$a_1 + a_2 + a_3 = n$$$. The third line of the input contains three integers $$$b_1, b_2, b_3$$$ ($$$0 \le b_j \le n$$$) — the number of times Bob will show rock, scissors and paper, respectively. It is guaranteed that $$$b_1 + b_2 + b_3 = n$$$.

Print two integers: the minimum and the maximum number of rounds Alice can win.

NoteIn the first example, Alice will not win any rounds if she shows scissors and then paper and Bob shows rock and then scissors. In the best outcome, Alice will win one round if she shows paper and then scissors, and Bob shows rock and then scissors.In the second example, Alice will not win any rounds if Bob shows the same things as Alice each round.In the third example, Alice always shows paper and Bob always shows rock so Alice will win all three rounds anyway.


Iahub and Iahubina went to a picnic in a forest full of trees. Less than 5 minutes passed before Iahub remembered of trees from programming. Moreover, he invented a new problem and Iahubina has to solve it, otherwise Iahub won't give her the food. Iahub asks Iahubina: can you build a rooted tree, such that  each internal node (a node with at least one son) has at least two sons;  node i has ci nodes in its subtree? Iahubina has to guess the tree. Being a smart girl, she realized that it's possible no tree can follow Iahub's restrictions. In this way, Iahub will eat all the food. You need to help Iahubina: determine if there's at least one tree following Iahub's restrictions. The required tree must contain n nodes.

The first line of the input contains integer n (1 ≤ n ≤ 24). Next line contains n positive integers: the i-th number represents ci (1 ≤ ci ≤ n).

Output on the first line "YES" (without quotes) if there exist at least one tree following Iahub's restrictions, otherwise output "NO" (without quotes).


Ksenia has her winter exams. Today she is learning combinatorics. Here's one of the problems she needs to learn to solve.How many distinct trees are there consisting of n vertices, each with the following properties:  the tree is marked, that is, the vertices of the tree are numbered from 1 to n;  each vertex of the tree is connected with at most three other vertices, and at the same moment the vertex with number 1 is connected with at most two other vertices;  the size of the tree's maximum matching equals k. Two trees are considered distinct if there are such two vertices u and v, that in one tree they are connected by an edge and in the other tree they are not.Help Ksenia solve the problem for the given n and k. As the answer to the problem can be very huge you should output it modulo 1000000007 (109 + 7).

The first line contains two integers n, k (1 ≤ n, k ≤ 50).

Print a single integer — the answer to the problem modulo 1000000007 (109 + 7).

NoteIf you aren't familiar with matchings, please, read the following link: http://en.wikipedia.org/wiki/Matching_(graph_theory).


A string a of length m is called antipalindromic iff m is even, and for each i (1 ≤ i ≤ m) ai ≠ am - i + 1.Ivan has a string s consisting of n lowercase Latin letters; n is even. He wants to form some string t that will be an antipalindromic permutation of s. Also Ivan has denoted the beauty of index i as bi, and the beauty of t as the sum of bi among all indices i such that si = ti.Help Ivan to determine maximum possible beauty of t he can get.

The first line contains one integer n (2 ≤ n ≤ 100, n is even) — the number of characters in s. The second line contains the string s itself. It consists of only lowercase Latin letters, and it is guaranteed that its letters can be reordered to form an antipalindromic string. The third line contains n integer numbers b1, b2, ..., bn (1 ≤ bi ≤ 100), where bi is the beauty of index i.

Print one number — the maximum possible beauty of t.


Ivan likes to learn different things about numbers, but he is especially interested in really big numbers. Ivan thinks that a positive integer number x is really big if the difference between x and the sum of its digits (in decimal representation) is not less than s. To prove that these numbers may have different special properties, he wants to know how rare (or not rare) they are — in fact, he needs to calculate the quantity of really big numbers that are not greater than n.Ivan tried to do the calculations himself, but soon realized that it's too difficult for him. So he asked you to help him in calculations.

The first (and the only) line contains two integers n and s (1 ≤ n, s ≤ 1018).

Print one integer — the quantity of really big numbers that are not greater than n.

NoteIn the first example numbers 10, 11 and 12 are really big.In the second example there are no really big numbers that are not greater than 25 (in fact, the first really big number is 30: 30 - 3 ≥ 20).In the third example 10 is the only really big number (10 - 1 ≥ 9).


Duff is in love with lovely numbers! A positive integer x is called lovely if and only if there is no such positive integer a > 1 such that a2 is a divisor of x.  Malek has a number store! In his store, he has only divisors of positive integer n (and he has all of them). As a birthday present, Malek wants to give her a lovely number from his store. He wants this number to be as big as possible.Malek always had issues in math, so he asked for your help. Please tell him what is the biggest lovely number in his store.

The first and only line of input contains one integer, n (1 ≤ n ≤ 1012).

NoteIn first sample case, there are numbers 1, 2, 5 and 10 in the shop. 10 isn't divisible by any perfect square, so 10 is lovely.In second sample case, there are numbers 1, 2, 3, 4, 6 and 12 in the shop. 12 is divisible by 4 = 22, so 12 is not lovely, while 6 is indeed lovely.


A string $$$s$$$ of length $$$n$$$ can be encrypted by the following algorithm:  iterate over all divisors of $$$n$$$ in decreasing order (i.e. from $$$n$$$ to $$$1$$$),  for each divisor $$$d$$$, reverse the substring $$$s[1 \dots d]$$$ (i.e. the substring which starts at position $$$1$$$ and ends at position $$$d$$$). For example, the above algorithm applied to the string $$$s$$$="codeforces" leads to the following changes: "codeforces" $$$\to$$$ "secrofedoc" $$$\to$$$ "orcesfedoc" $$$\to$$$ "rocesfedoc" $$$\to$$$ "rocesfedoc" (obviously, the last reverse operation doesn't change the string because $$$d=1$$$).You are given the encrypted string $$$t$$$. Your task is to decrypt this string, i.e., to find a string $$$s$$$ such that the above algorithm results in string $$$t$$$. It can be proven that this string $$$s$$$ always exists and is unique.

The first line of input consists of a single integer $$$n$$$ ($$$1 \le n \le 100$$$) — the length of the string $$$t$$$. The second line of input consists of the string $$$t$$$. The length of $$$t$$$ is $$$n$$$, and it consists only of lowercase Latin letters.

Print a string $$$s$$$ such that the above algorithm results in $$$t$$$.

NoteThe first example is described in the problem statement.


The Tower of Hanoi is a well-known mathematical puzzle. It consists of three rods, and a number of disks of different sizes which can slide onto any rod. The puzzle starts with the disks in a neat stack in ascending order of size on one rod, the smallest at the top, thus making a conical shape.The objective of the puzzle is to move the entire stack to another rod, obeying the following simple rules:   Only one disk can be moved at a time.  Each move consists of taking the upper disk from one of the stacks and placing it on top of another stack i.e. a disk can only be moved if it is the uppermost disk on a stack.  No disk may be placed on top of a smaller disk. With three disks, the puzzle can be solved in seven moves. The minimum number of moves required to solve a Tower of Hanoi puzzle is 2n - 1, where n is the number of disks. (c) Wikipedia.SmallY's puzzle is very similar to the famous Tower of Hanoi. In the Tower of Hanoi puzzle you need to solve a puzzle in minimum number of moves, in SmallY's puzzle each move costs some money and you need to solve the same puzzle but for minimal cost. At the beginning of SmallY's puzzle all n disks are on the first rod. Moving a disk from rod i to rod j (1 ≤ i, j ≤ 3) costs tij units of money. The goal of the puzzle is to move all the disks to the third rod.In the problem you are given matrix t and an integer n. You need to count the minimal cost of solving SmallY's puzzle, consisting of n disks.

Each of the first three lines contains three integers — matrix t. The j-th integer in the i-th line is tij (1 ≤ tij ≤ 10000; i ≠ j). The following line contains a single integer n (1 ≤ n ≤ 40) — the number of disks. It is guaranteed that for all i (1 ≤ i ≤ 3), tii = 0.

Print a single integer — the minimum cost of solving SmallY's puzzle.


Vasya is very upset that many people on the Net mix uppercase and lowercase letters in one word. That's why he decided to invent an extension for his favorite browser that would change the letters' register in every word so that it either only consisted of lowercase letters or, vice versa, only of uppercase ones. At that as little as possible letters should be changed in the word. For example, the word HoUse must be replaced with house, and the word ViP — with VIP. If a word contains an equal number of uppercase and lowercase letters, you should replace all the letters with lowercase ones. For example, maTRIx should be replaced by matrix. Your task is to use the given method on one given word.

The first line contains a word s — it consists of uppercase and lowercase Latin letters and possesses the length from 1 to 100.

Print the corrected word s. If the given word s has strictly more uppercase letters, make the word written in the uppercase register, otherwise - in the lowercase one.


You are given the current time in 24-hour format hh:mm. Find and print the time after a minutes.Note that you should find only the time after a minutes, see the examples to clarify the problem statement.You can read more about 24-hour format here https://en.wikipedia.org/wiki/24-hour_clock.

The first line contains the current time in the format hh:mm (0 ≤ hh < 24, 0 ≤ mm < 60). The hours and the minutes are given with two digits (the hours or the minutes less than 10 are given with the leading zeroes). The second line contains integer a (0 ≤ a ≤ 104) — the number of the minutes passed.

The only line should contain the time after a minutes in the format described in the input. Note that you should print exactly two digits for the hours and the minutes (add leading zeroes to the numbers if needed). See the examples to check the input/output format.


The football season has just ended in Berland. According to the rules of Berland football, each match is played between two teams. The result of each match is either a draw, or a victory of one of the playing teams. If a team wins the match, it gets $$$w$$$ points, and the opposing team gets $$$0$$$ points. If the game results in a draw, both teams get $$$d$$$ points.The manager of the Berland capital team wants to summarize the results of the season, but, unfortunately, all information about the results of each match is lost. The manager only knows that the team has played $$$n$$$ games and got $$$p$$$ points for them.You have to determine three integers $$$x$$$, $$$y$$$ and $$$z$$$ — the number of wins, draws and loses of the team. If there are multiple answers, print any of them. If there is no suitable triple $$$(x, y, z)$$$, report about it.

The first line contains four integers $$$n$$$, $$$p$$$, $$$w$$$ and $$$d$$$ $$$(1 \le n \le 10^{12}, 0 \le p \le 10^{17}, 1 \le d < w \le 10^{5})$$$ — the number of games, the number of points the team got, the number of points awarded for winning a match, and the number of points awarded for a draw, respectively. Note that $$$w > d$$$, so the number of points awarded for winning is strictly greater than the number of points awarded for draw.

If there is no answer, print $$$-1$$$. Otherwise print three non-negative integers $$$x$$$, $$$y$$$ and $$$z$$$ — the number of wins, draws and losses of the team. If there are multiple possible triples $$$(x, y, z)$$$, print any of them. The numbers should meet the following conditions:    $$$x \cdot w + y \cdot d = p$$$,  $$$x + y + z = n$$$.

NoteOne of the possible answers in the first example — $$$17$$$ wins, $$$9$$$ draws and $$$4$$$ losses. Then the team got $$$17 \cdot 3 + 9 \cdot 1 = 60$$$ points in $$$17 + 9 + 4 = 30$$$ games.In the second example the maximum possible score is $$$10 \cdot 5 = 50$$$. Since $$$p = 51$$$, there is no answer.In the third example the team got $$$0$$$ points, so all $$$20$$$ games were lost.


Есть n-подъездный дом, в каждом подъезде по m этажей, и на каждом этаже каждого подъезда ровно k квартир. Таким образом, в доме всего n·m·k квартир. Они пронумерованы естественным образом от 1 до n·m·k, то есть первая квартира на первом этаже в первом подъезде имеет номер 1, первая квартира на втором этаже первого подъезда имеет номер k + 1 и так далее. Особенность этого дома состоит в том, что он круглый. То есть если обходить его по часовой стрелке, то после подъезда номер 1 следует подъезд номер 2, затем подъезд номер 3 и так далее до подъезда номер n. После подъезда номер n снова идёт подъезд номер 1.Эдвард живёт в квартире номер a, а Наташа — в квартире номер b. Переход на 1 этаж вверх или вниз по лестнице занимает 5 секунд, переход от двери подъезда к двери соседнего подъезда — 15 секунд, а переход в пределах одного этажа одного подъезда происходит мгновенно. Также в каждом подъезде дома есть лифт. Он устроен следующим образом: он всегда приезжает ровно через 10 секунд после вызова, а чтобы переместить пассажира на один этаж вверх или вниз, лифт тратит ровно 1 секунду. Посадка и высадка происходят мгновенно.Помогите Эдварду найти минимальное время, за которое он сможет добраться до квартиры Наташи. Считайте, что Эдвард может выйти из подъезда только с первого этажа соответствующего подъезда (это происходит мгновенно). Если Эдвард стоит перед дверью какого-то подъезда, он может зайти в него и сразу окажется на первом этаже этого подъезда (это также происходит мгновенно). Эдвард может выбирать, в каком направлении идти вокруг дома.

В первой строке входных данных следуют три числа n, m, k (1 ≤ n, m, k ≤ 1000) — количество подъездов в доме, количество этажей в каждом подъезде и количество квартир на каждом этаже каждого подъезда соответственно. Во второй строке входных данных записаны два числа a и b (1 ≤ a, b ≤ n·m·k) — номера квартир, в которых живут Эдвард и Наташа, соответственно. Гарантируется, что эти номера различны.

Выведите единственное целое число — минимальное время (в секундах), за которое Эдвард сможет добраться от своей квартиры до квартиры Наташи.

ПримечаниеВ первом тестовом примере Эдвард находится в 4 подъезде на 10 этаже, а Наташа находится в 1 подъезде на 2 этаже. Поэтому Эдварду выгодно сначала спуститься на лифте на первый этаж (на это он потратит 19 секунд, из которых 10 — на ожидание и 9 — на поездку на лифте), затем обойти дом против часовой стрелки до подъезда номер 1 (на это он потратит 15 секунд), и наконец подняться по лестнице на этаж номер 2 (на это он потратит 5 секунд). Таким образом, ответ равен 19 + 15 + 5 = 39.Во втором тестовом примере Эдвард живёт в подъезде 2 на этаже 1, а Наташа находится в подъезде 1 на этаже 1. Поэтому Эдварду выгодно просто обойти дом по часовой стрелке до подъезда 1, на это он потратит 15 секунд.


Rock... Paper!After Karen have found the deterministic winning (losing?) strategy for rock-paper-scissors, her brother, Koyomi, comes up with a new game as a substitute. The game works as follows.A positive integer n is decided first. Both Koyomi and Karen independently choose n distinct positive integers, denoted by x1, x2, ..., xn and y1, y2, ..., yn respectively. They reveal their sequences, and repeat until all of 2n integers become distinct, which is the only final state to be kept and considered.Then they count the number of ordered pairs (i, j) (1 ≤ i, j ≤ n) such that the value xi xor yj equals to one of the 2n integers. Here xor means the bitwise exclusive or operation on two integers, and is denoted by operators ^ and/or xor in most programming languages.Karen claims a win if the number of such pairs is even, and Koyomi does otherwise. And you're here to help determine the winner of their latest game.

The first line of input contains a positive integer n (1 ≤ n ≤ 2 000) — the length of both sequences. The second line contains n space-separated integers x1, x2, ..., xn (1 ≤ xi ≤ 2·106) — the integers finally chosen by Koyomi. The third line contains n space-separated integers y1, y2, ..., yn (1 ≤ yi ≤ 2·106) — the integers finally chosen by Karen. Input guarantees that the given 2n integers are pairwise distinct, that is, no pair (i, j) (1 ≤ i, j ≤ n) exists such that one of the following holds: xi = yj; i ≠ j and xi = xj; i ≠ j and yi = yj.

Output one line — the name of the winner, that is, "Koyomi" or "Karen" (without quotes). Please be aware of the capitalization.

NoteIn the first example, there are 6 pairs satisfying the constraint: (1, 1), (1, 2), (2, 1), (2, 3), (3, 2) and (3, 3). Thus, Karen wins since 6 is an even number.In the second example, there are 16 such pairs, and Karen wins again.


Right now she actually isn't. But she will be, if you don't solve this problem.You are given integers n, k, A and B. There is a number x, which is initially equal to n. You are allowed to perform two types of operations:   Subtract 1 from x. This operation costs you A coins.  Divide x by k. Can be performed only if x is divisible by k. This operation costs you B coins.  What is the minimum amount of coins you have to pay to make x equal to 1?

The first line contains a single integer n (1 ≤ n ≤ 2·109). The second line contains a single integer k (1 ≤ k ≤ 2·109). The third line contains a single integer A (1 ≤ A ≤ 2·109). The fourth line contains a single integer B (1 ≤ B ≤ 2·109).

Output a single integer — the minimum amount of coins you have to pay to make x equal to 1.

NoteIn the first testcase, the optimal strategy is as follows:   Subtract 1 from x (9 → 8) paying 3 coins.  Divide x by 2 (8 → 4) paying 1 coin.  Divide x by 2 (4 → 2) paying 1 coin.  Divide x by 2 (2 → 1) paying 1 coin. The total cost is 6 coins.In the second test case the optimal strategy is to subtract 1 from x 4 times paying 8 coins in total.


Today is the final contest of INOI (Iranian National Olympiad in Informatics). The contest room is a row with $$$n$$$ computers. All computers are numbered with integers from $$$1$$$ to $$$n$$$ from left to right. There are $$$m$$$ participants, numbered with integers from $$$1$$$ to $$$m$$$.We have an array $$$a$$$ of length $$$m$$$ where $$$a_{i}$$$ ($$$1 \leq a_i \leq n$$$) is the computer behind which the $$$i$$$-th participant wants to sit.Also, we have another array $$$b$$$ of length $$$m$$$ consisting of characters 'L' and 'R'. $$$b_i$$$ is the side from which the $$$i$$$-th participant enters the room. 'L' means the participant enters from the left of computer $$$1$$$ and goes from left to right, and 'R' means the participant enters from the right of computer $$$n$$$ and goes from right to left.The participants in the order from $$$1$$$ to $$$m$$$ enter the room one by one. The $$$i$$$-th of them enters the contest room in the direction $$$b_i$$$ and goes to sit behind the $$$a_i$$$-th computer. If it is occupied he keeps walking in his direction until he reaches the first unoccupied computer. After that, he sits behind it. If he doesn't find any computer he gets upset and gives up on the contest.The madness of the $$$i$$$-th participant is the distance between his assigned computer ($$$a_i$$$) and the computer he ends up sitting behind. The distance between computers $$$i$$$ and $$$j$$$ is equal to $$$|i - j|$$$.The values in the array $$$a$$$ can be equal. There exist $$$n^m \cdot 2^m$$$ possible pairs of arrays $$$(a, b)$$$.Consider all pairs of arrays $$$(a, b)$$$ such that no person becomes upset. For each of them let's calculate the sum of participants madnesses. Find the sum of all these values.You will be given some prime modulo $$$p$$$. Find this sum by modulo $$$p$$$.

The only line contains three integers $$$n$$$, $$$m$$$, $$$p$$$ ($$$1 \leq m \leq n \leq 500, 10^8 \leq p \leq 10 ^ 9 + 9$$$). It is guaranteed, that the number $$$p$$$ is prime.

Print only one integer — the required sum by modulo $$$p$$$.

NoteIn the first test, there are three possible arrays $$$a$$$: $$$\{1\}$$$, $$$\{2\}$$$, and $$$ \{3\}$$$ and two possible arrays $$$b$$$: $$$\{\mathtt{L}\}$$$ and $$$\{\mathtt{R}\}$$$. For all six pairs of arrays $$$(a, b)$$$, the only participant will sit behind the computer $$$a_1$$$, so his madness will be $$$0$$$. So the total sum of madnesses will be $$$0$$$.In the second test, all possible pairs of arrays $$$(a, b)$$$, such that no person becomes upset are:  $$$(\{1, 1\}, \{\mathtt{L}, \mathtt{L}\})$$$, the sum of madnesses is $$$1$$$;  $$$(\{1, 1\}, \{\mathtt{R}, \mathtt{L}\})$$$, the sum of madnesses is $$$1$$$;  $$$(\{2, 2\}, \{\mathtt{R}, \mathtt{R}\})$$$, the sum of madnesses is $$$1$$$;  $$$(\{2, 2\}, \{\mathtt{L}, \mathtt{R}\})$$$, the sum of madnesses is $$$1$$$;  all possible pairs of $$$a \in \{\{1, 2\}, \{2, 1\}\}$$$ and $$$b \in \{\{\mathtt{L}, \mathtt{L}\}, \{\mathtt{R}, \mathtt{L}\}, \{\mathtt{L}, \mathtt{R}\}, \{\mathtt{R}, \mathtt{R}\}\}$$$, the sum of madnesses is $$$0$$$. So, the answer is $$$1 + 1 + 1 + 1 + 0 \ldots = 4$$$.


You have two positive integers w and h. Your task is to count the number of rhombi which have the following properties:   Have positive area.  With vertices at integer points.  All vertices of the rhombi are located inside or on the border of the rectangle with vertices at points (0, 0), (w, 0), (w, h), (0, h). In other words, for all vertices (xi, yi) of the rhombus the following conditions should fulfill: 0 ≤ xi ≤ w and 0 ≤ yi ≤ h.  Its diagonals are parallel to the axis.  Count the number of such rhombi.Let us remind you that a rhombus is a quadrilateral whose four sides all have the same length.

The first line contains two integers w and h (1 ≤ w, h ≤ 4000) — the rectangle's sizes.

Print a single number — the number of sought rhombi. Please do not use the %lld specifier to read or write 64-bit integers in С++. It is preferred to use cin, cout streams or the %I64d specifier.

NoteIn the first example there exists only one such rhombus. Its vertices are located at points (1, 0), (2, 1), (1, 2), (0, 1).


Artem has an array of n positive integers. Artem decided to play with it. The game consists of n moves. Each move goes like this. Artem chooses some element of the array and removes it. For that, he gets min(a, b) points, where a and b are numbers that were adjacent with the removed number. If the number doesn't have an adjacent number to the left or right, Artem doesn't get any points. After the element is removed, the two parts of the array glue together resulting in the new array that Artem continues playing with. Borya wondered what maximum total number of points Artem can get as he plays this game.

The first line contains a single integer n (1 ≤ n ≤ 5·105) — the number of elements in the array. The next line contains n integers ai (1 ≤ ai ≤ 106) — the values of the array elements.

In a single line print a single integer — the maximum number of points Artem can get.


Little Elephant loves magic squares very much.A magic square is a 3 × 3 table, each cell contains some positive integer. At that the sums of integers in all rows, columns and diagonals of the table are equal. The figure below shows the magic square, the sum of integers in all its rows, columns and diagonals equals 15.  The Little Elephant remembered one magic square. He started writing this square on a piece of paper, but as he wrote, he forgot all three elements of the main diagonal of the magic square. Fortunately, the Little Elephant clearly remembered that all elements of the magic square did not exceed 105. Help the Little Elephant, restore the original magic square, given the Elephant's notes.

The first three lines of the input contain the Little Elephant's notes. The first line contains elements of the first row of the magic square. The second line contains the elements of the second row, the third line is for the third row. The main diagonal elements that have been forgotten by the Elephant are represented by zeroes. It is guaranteed that the notes contain exactly three zeroes and they are all located on the main diagonal. It is guaranteed that all positive numbers in the table do not exceed 105.

Print three lines, in each line print three integers — the Little Elephant's magic square. If there are multiple magic squares, you are allowed to print any of them. Note that all numbers you print must be positive and not exceed 105. It is guaranteed that there exists at least one magic square that meets the conditions.


Little Johnny Bubbles enjoys spending hours in front of his computer playing video games. His favorite game is Bubble Strike, fast-paced bubble shooting online game for two players.Each game is set in one of the N maps, each having different terrain configuration. First phase of each game decides on which map the game will be played. The game system randomly selects three maps and shows them to the players. Each player must pick one of those three maps to be discarded. The game system then randomly selects one of the maps that were not picked by any of the players and starts the game.Johnny is deeply enthusiastic about the game and wants to spend some time studying maps, thus increasing chances to win games played on those maps. However, he also needs to do his homework, so he does not have time to study all the maps. That is why he asked himself the following question: "What is the minimum number of maps I have to study, so that the probability to play one of those maps is at least $$$P$$$"?Can you help Johnny find the answer for this question? You can assume Johnny's opponents do not know him, and they will randomly pick maps.

The first line contains two integers $$$N$$$ ($$$3$$$ $$$\leq$$$ $$$N$$$ $$$\leq$$$ $$$10^{3}$$$) and $$$P$$$ ($$$0$$$ $$$\leq$$$ $$$P$$$ $$$\leq$$$ $$$1$$$) – total number of maps in the game and probability to play map Johnny has studied. $$$P$$$ will have at most four digits after the decimal point.

Output contains one integer number – minimum number of maps Johnny has to study.


You have a nuts and lots of boxes. The boxes have a wonderful feature: if you put x (x ≥ 0) divisors (the spacial bars that can divide a box) to it, you get a box, divided into x + 1 sections.You are minimalist. Therefore, on the one hand, you are against dividing some box into more than k sections. On the other hand, you are against putting more than v nuts into some section of the box. What is the minimum number of boxes you have to use if you want to put all the nuts in boxes, and you have b divisors?Please note that you need to minimize the number of used boxes, not sections. You do not have to minimize the number of used divisors.

The first line contains four space-separated integers k, a, b, v (2 ≤ k ≤ 1000; 1 ≤ a, b, v ≤ 1000) — the maximum number of sections in the box, the number of nuts, the number of divisors and the capacity of each section of the box.

Print a single integer — the answer to the problem.

NoteIn the first sample you can act like this:   Put two divisors to the first box. Now the first box has three sections and we can put three nuts into each section. Overall, the first box will have nine nuts.  Do not put any divisors into the second box. Thus, the second box has one section for the last nut. In the end we've put all the ten nuts into boxes.The second sample is different as we have exactly one divisor and we put it to the first box. The next two boxes will have one section each.


— This is not playing but duty as allies of justice, Nii-chan!— Not allies but justice itself, Onii-chan!With hands joined, go everywhere at a speed faster than our thoughts! This time, the Fire Sisters — Karen and Tsukihi — is heading for somewhere they've never reached — water-surrounded islands!There are three clusters of islands, conveniently coloured red, blue and purple. The clusters consist of a, b and c distinct islands respectively.Bridges have been built between some (possibly all or none) of the islands. A bridge bidirectionally connects two different islands and has length 1. For any two islands of the same colour, either they shouldn't be reached from each other through bridges, or the shortest distance between them is at least 3, apparently in order to prevent oddities from spreading quickly inside a cluster.The Fire Sisters are ready for the unknown, but they'd also like to test your courage. And you're here to figure out the number of different ways to build all bridges under the constraints, and give the answer modulo 998 244 353. Two ways are considered different if a pair of islands exist, such that there's a bridge between them in one of them, but not in the other.

The first and only line of input contains three space-separated integers a, b and c (1 ≤ a, b, c ≤ 5 000) — the number of islands in the red, blue and purple clusters, respectively.

Output one line containing an integer — the number of different ways to build bridges, modulo 998 244 353.

NoteIn the first example, there are 3 bridges that can possibly be built, and no setup of bridges violates the restrictions. Thus the answer is 23 = 8.In the second example, the upper two structures in the figure below are instances of valid ones, while the lower two are invalid due to the blue and purple clusters, respectively.


A Thue-Morse-Radecki-Mateusz sequence (Thorse-Radewoosh sequence in short) is an infinite sequence constructed from a finite sequence $$$\mathrm{gen}$$$ of length $$$d$$$ and an integer $$$m$$$, obtained in the following sequence of steps:  In the beginning, we define the one-element sequence $$$M_0=(0)$$$. In the $$$k$$$-th step, $$$k \geq 1$$$, we define the sequence $$$M_k$$$ to be the concatenation of the $$$d$$$ copies of $$$M_{k-1}$$$. However, each of them is altered slightly — in the $$$i$$$-th of them ($$$1 \leq i \leq d$$$), each element $$$x$$$ is changed to $$$(x+\mathrm{gen}_i) \pmod{m}$$$. For instance, if we pick $$$\mathrm{gen} = (0, \color{blue}{1}, \color{green}{2})$$$ and $$$m = 4$$$:   $$$M_0 = (0)$$$,  $$$M_1 = (0, \color{blue}{1}, \color{green}{2})$$$,  $$$M_2 = (0, 1, 2, \color{blue}{1, 2, 3}, \color{green}{2, 3, 0})$$$,  $$$M_3 = (0, 1, 2, 1, 2, 3, 2, 3, 0, \color{blue}{1, 2, 3, 2, 3, 0, 3, 0, 1}, \color{green}{2, 3, 0, 3, 0, 1, 0, 1, 2})$$$, and so on. As you can see, as long as the first element of $$$\mathrm{gen}$$$ is $$$0$$$, each consecutive step produces a sequence whose prefix is the sequence generated in the previous step. Therefore, we can define the infinite Thorse-Radewoosh sequence $$$M_\infty$$$ as the sequence obtained by applying the step above indefinitely. For the parameters above, $$$M_\infty = (0, 1, 2, 1, 2, 3, 2, 3, 0, 1, 2, 3, 2, 3, 0, 3, 0, 1, \dots)$$$.Mateusz picked a sequence $$$\mathrm{gen}$$$ and an integer $$$m$$$, and used them to obtain a Thorse-Radewoosh sequence $$$M_\infty$$$. He then picked two integers $$$l$$$, $$$r$$$, and wrote down a subsequence of this sequence $$$A := ((M_\infty)_l, (M_\infty)_{l+1}, \dots, (M_\infty)_r)$$$.Note that we use the $$$1$$$-based indexing both for $$$M_\infty$$$ and $$$A$$$.Mateusz has his favorite sequence $$$B$$$ with length $$$n$$$, and would like to see how large it is compared to $$$A$$$. Let's say that $$$B$$$ majorizes sequence $$$X$$$ of length $$$n$$$ (let's denote it as $$$B \geq X$$$) if and only if for all $$$i \in \{1, 2, \dots, n\}$$$, we have $$$B_i \geq X_i$$$.He now asks himself how many integers $$$x$$$ in the range $$$[1, |A| - n + 1]$$$ there are such that $$$B \geq (A_x, A_{x+1}, A_{x+2}, \dots, A_{x+n-1})$$$. As both sequences were huge, answering the question using only his pen and paper turned out to be too time-consuming. Can you help him automate his research?

The first line contains two integers $$$d$$$ and $$$m$$$ ($$$2 \leq d \leq 20$$$, $$$2 \leq m \leq 60$$$) — the length of the sequence $$$\mathrm{gen}$$$ and an integer used to perform the modular operations. The second line contains $$$d$$$ integers $$$\mathrm{gen}_i$$$ ($$$0 \leq \mathrm{gen}_i < m$$$). It's guaranteed that the first element of the sequence $$$\mathrm{gen}$$$ is equal to zero. The third line contains one integer $$$n$$$ ($$$1 \leq n \leq 30000$$$) — the length of the sequence $$$B$$$. The fourth line contains $$$n$$$ integers $$$B_i$$$ ($$$0 \leq B_i < m$$$). The fifth line contains two integers $$$l$$$ and $$$r$$$ ($$$1 \leq l \leq r \leq 10^{18}$$$, $$$r-l+1 \geq n$$$).

NoteThorse-Radewoosh sequence in the first example is the standard Thue-Morse sequence, so the sequence $$$A$$$ is as follows: $$$11010011001011010010$$$. Here are the places where the sequence $$$B$$$ majorizes $$$A$$$:


Alexander is learning how to convert numbers from the decimal system to any other, however, he doesn't know English letters, so he writes any number only as a decimal number, it means that instead of the letter A he will write the number 10. Thus, by converting the number 475 from decimal to hexadecimal system, he gets 11311 (475 = 1·162 + 13·161 + 11·160). Alexander lived calmly until he tried to convert the number back to the decimal number system.Alexander remembers that he worked with little numbers so he asks to find the minimum decimal number so that by converting it to the system with the base n he will get the number k.

The first line contains the integer n (2 ≤ n ≤ 109). The second line contains the integer k (0 ≤ k < 1060), it is guaranteed that the number k contains no more than 60 symbols. All digits in the second line are strictly less than n. Alexander guarantees that the answer exists and does not exceed 1018. The number k doesn't contain leading zeros.

Print the number x (0 ≤ x ≤ 1018) — the answer to the problem.

NoteIn the first example 12 could be obtained by converting two numbers to the system with base 13: 12 = 12·130 or 15 = 1·131 + 2·130.


You are given n rectangles. The corners of rectangles have integer coordinates and their edges are parallel to the Ox and Oy axes. The rectangles may touch each other, but they do not overlap (that is, there are no points that belong to the interior of more than one rectangle). Your task is to determine if the rectangles form a square. In other words, determine if the set of points inside or on the border of at least one rectangle is precisely equal to the set of points inside or on the border of some square.

The first line contains a single integer n (1 ≤ n ≤ 5). Next n lines contain four integers each, describing a single rectangle: x1, y1, x2, y2 (0 ≤ x1 < x2 ≤ 31400, 0 ≤ y1 < y2 ≤ 31400) — x1 and x2 are x-coordinates of the left and right edges of the rectangle, and y1 and y2 are y-coordinates of the bottom and top edges of the rectangle.  No two rectangles overlap (that is, there are no points that belong to the interior of more than one rectangle).

In a single line print "YES", if the given rectangles form a square, or "NO" otherwise.


Monocarp has decided to buy a new TV set and hang it on the wall in his flat. The wall has enough free space so Monocarp can buy a TV set with screen width not greater than $$$a$$$ and screen height not greater than $$$b$$$. Monocarp is also used to TV sets with a certain aspect ratio: formally, if the width of the screen is $$$w$$$, and the height of the screen is $$$h$$$, then the following condition should be met: $$$\frac{w}{h} = \frac{x}{y}$$$.There are many different TV sets in the shop. Monocarp is sure that for any pair of positive integers $$$w$$$ and $$$h$$$ there is a TV set with screen width $$$w$$$ and height $$$h$$$ in the shop.Monocarp isn't ready to choose the exact TV set he is going to buy. Firstly he wants to determine the optimal screen resolution. He has decided to try all possible variants of screen size. But he must count the number of pairs of positive integers $$$w$$$ and $$$h$$$, beforehand, such that $$$(w \le a)$$$, $$$(h \le b)$$$ and $$$(\frac{w}{h} = \frac{x}{y})$$$.In other words, Monocarp wants to determine the number of TV sets having aspect ratio $$$\frac{x}{y}$$$, screen width not exceeding $$$a$$$, and screen height not exceeding $$$b$$$. Two TV sets are considered different if they have different screen width or different screen height.

The first line contains four integers $$$a$$$, $$$b$$$, $$$x$$$, $$$y$$$ ($$$1 \le a, b, x, y \le 10^{18}$$$) — the constraints on the screen width and height, and on the aspect ratio.

Print one integer — the number of different variants to choose TV screen width and screen height so that they meet the aforementioned constraints.

NoteIn the first example, there are $$$3$$$ possible variants: $$$(5, 3)$$$, $$$(10, 6)$$$, $$$(15, 9)$$$.In the second example, there is no TV set meeting the constraints.In the third example, there is only one variant: $$$(3, 2)$$$.


Let's analyze a program written on some strange programming language. The variables in this language have names consisting of $$$1$$$ to $$$4$$$ characters, and each character is a lowercase or an uppercase Latin letter, or a digit. There is an extra constraint that the first character should not be a digit.There are four types of operations in the program, each denoted by one of the characters: $, ^, # or &.Each line of the program has one of the following formats:   <lvalue>=<rvalue>, where <lvalue> and <rvalue> are valid variable names;  <lvalue>=<arg1><op><arg2>, where <lvalue>, <arg1> and <arg2> are valid variable names, and <op> is an operation character. The program is executed line-by-line, and the result of execution is stored in a variable having the name res. If res is never assigned in the program, then the result will be equal to the value of res before running the program.Two programs are called equivalent if no matter which operations do characters $, ^, # and & denote (but, obviously, performing the same operation on the same arguments gives the same result) and which values do variables have before execution of program, the value of res after running the first program is equal to the value of res after running the second program (the programs are executed independently).You are given a program consisting of $$$n$$$ lines. Your task is to write a program consisting of minimum possible number of lines that is equivalent to the program you are given.

The first line contains one integer $$$n$$$ ($$$1 \le n \le 1000$$$) — the number of lines in the program. Then $$$n$$$ lines follow — the program itself. Each line corresponds to the format described in the statement and has no extra whitespaces.

In the first line print $$$k$$$ — the minimum number of lines in the equivalent program. Then print $$$k$$$ lines without any whitespaces — an equivalent program having exactly $$$k$$$ lines, in the same format it is described in the statement.


You won't find this sequence on OEIS.


Do you remember how Kai constructed the word "eternity" using pieces of ice as components?Little Sheldon plays with pieces of ice, each piece has exactly one digit between 0 and 9. He wants to construct his favourite number t. He realized that digits 6 and 9 are very similar, so he can rotate piece of ice with 6 to use as 9 (and vice versa). Similary, 2 and 5 work the same. There is no other pair of digits with similar effect. He called this effect "Digital Mimicry".Sheldon favourite number is t. He wants to have as many instances of t as possible. How many instances he can construct using the given sequence of ice pieces. He can use any piece at most once.

The first line contains integer t (1 ≤ t ≤ 10000). The second line contains the sequence of digits on the pieces. The length of line is equal to the number of pieces and between 1 and 200, inclusive. It contains digits between 0 and 9.

NoteThis problem contains very weak pretests.


Harry Potter lost his Invisibility Cloak, running from the school caretaker Filch. Finding an invisible object is not an easy task. Fortunately, Harry has friends who are willing to help. Hermione Granger had read "The Invisibility Cloaks, and Everything about Them", as well as six volumes of "The Encyclopedia of Quick Search of Shortest Paths in Graphs, Network Flows, the Maximal Increasing Subsequences and Other Magical Objects". She has already developed a search algorithm for the invisibility cloak in complex dynamic systems (Hogwarts is one of them).Hogwarts consists of n floors, numbered by integers from 1 to n. Some pairs of floors are connected by staircases. The staircases may change its position, moving exactly one end. Formally the situation is like this: if a staircase connects the floors a and b, then in one move it may modify its position so as to connect the floors a and c or b and c, where c is any floor different from a and b. Under no circumstances the staircase can connect a floor with itself. At the same time there can be multiple stairs between a pair of floors.Initially, Harry is on the floor with the number 1. He does not remember on what floor he has lost the cloak and wants to look for it on each of the floors. Therefore, his goal is to visit each of n floors at least once. Harry can visit the floors in any order and finish the searching at any floor.Nowadays the staircases move quite rarely. However, Ron and Hermione are willing to put a spell on any of them to help Harry find the cloak. To cause less suspicion, the three friends plan to move the staircases one by one, and no more than once for each staircase. In between shifting the staircases Harry will be able to move about the floors, reachable at the moment from the staircases, and look for his Invisibility Cloak. It is assumed that during all this time the staircases will not move spontaneously.Help the three friends to compose a searching plan. If there are several variants to solve the problem, any valid option (not necessarily the optimal one) will be accepted.

The first line contains integers n and m (1 ≤ n ≤ 100000, 0 ≤ m ≤ 200000), which are the number of floors and staircases in Hogwarts, respectively. The following m lines contain pairs of floors connected by staircases at the initial moment of time.

In the first line print "YES" (without the quotes) if Harry is able to search all the floors, and "NO" otherwise. If the answer is positive, then print on the second line the number of staircases that Ron and Hermione will have to shift. Further output should look like this: Harry's moves  a staircase's move  Harry's moves a staircase's move ... a staircase's move Harry's moves Each "Harry's move" should be represented as a list of floors in the order in which they have been visited. The total amount of elements of these lists must not exceed 106. When you print each list, first print the number of elements in it, and then in the same line print the actual space-separated elements. The first number in the first list should be the number 1 (the floor, from which Harry begins to search). Any list except the first one might contain the zero number of elements. Note that Harry can visit some floors again, but must visit all n floors at least once. Two consecutively visited floors must be directly connected by a staircase (at the time Harry goes from one of them to the other one). No two floors that are visited consequtively can be equal. In the description of a "staircase's move" indicate the number of staircase (the staircases are numbered from 1 to m in the order in which they are given in the input data) and its new location (two numbers of the connected floors in any order). Any staircase can be moved at most once. If there are several solutions, output any.


Andrey received a postcard from Irina. It contained only the words "Hello, Andrey!", and a strange string consisting of lowercase Latin letters, snowflakes and candy canes. Andrey thought that this string is an encrypted message, and decided to decrypt it.Andrey noticed that snowflakes and candy canes always stand after the letters, so he supposed that the message was encrypted as follows. Candy cane means that the letter before it can be removed, or can be left. A snowflake means that the letter before it can be removed, left, or repeated several times.For example, consider the following string:   This string can encode the message «happynewyear». For this, candy canes and snowflakes should be used as follows:   candy cane 1: remove the letter w,  snowflake 1: repeat the letter p twice,  candy cane 2: leave the letter n,  snowflake 2: remove the letter w,  snowflake 3: leave the letter e.   Please note that the same string can encode different messages. For example, the string above can encode «hayewyar», «happpppynewwwwwyear», and other messages.Andrey knows that messages from Irina usually have a length of $$$k$$$ letters. Help him to find out if a given string can encode a message of $$$k$$$ letters, and if so, give an example of such a message.

The first line contains the string received in the postcard. The string consists only of lowercase Latin letters, as well as the characters «*» and «?», meaning snowflake and candy cone, respectively. These characters can only appear immediately after the letter. The length of the string does not exceed $$$200$$$. The second line contains an integer number $$$k$$$ ($$$1 \leq k \leq 200$$$), the required message length.

Print any message of length $$$k$$$ that the given string can encode, or «Impossible» if such a message does not exist.


You are solving the crossword problem K from IPSC 2014. You solved all the clues except for one: who does Eevee evolve into? You are not very into pokemons, but quick googling helped you find out, that Eevee can evolve into eight different pokemons: Vaporeon, Jolteon, Flareon, Espeon, Umbreon, Leafeon, Glaceon, and Sylveon.You know the length of the word in the crossword, and you already know some letters. Designers of the crossword made sure that the answer is unambiguous, so you can assume that exactly one pokemon out of the 8 that Eevee evolves into fits the length and the letters given. Your task is to find it.

First line contains an integer n (6 ≤ n ≤ 8) – the length of the string. Next line contains a string consisting of n characters, each of which is either a lower case english letter (indicating a known letter) or a dot character (indicating an empty cell in the crossword).

Print a name of the pokemon that Eevee can evolve into that matches the pattern in the input. Use lower case letters only to print the name (in particular, do not capitalize the first letter).

NoteHere's a set of names in a form you can paste into your solution:["vaporeon", "jolteon", "flareon", "espeon", "umbreon", "leafeon", "glaceon", "sylveon"]{"vaporeon", "jolteon", "flareon", "espeon", "umbreon", "leafeon", "glaceon", "sylveon"}


In Berland a money reform is being prepared. New coins are being introduced. After long economic calculations was decided that the most expensive coin should possess the denomination of exactly n Berland dollars. Also the following restriction has been introduced for comfort: the denomination of each coin should be divisible by the denomination of any cheaper coin. It is known that among all the possible variants the variant with the largest number of new coins will be chosen. Find this variant. Print in the order of decreasing of the coins' denominations.

The first and only line contains an integer n (1 ≤ n ≤ 106) which represents the denomination of the most expensive coin.

Print the denominations of all the coins in the order of decreasing. The number of coins must be the largest possible (with the given denomination n of the most expensive coin). Also, the denomination of every coin must be divisible by the denomination of any cheaper coin. Naturally, the denominations of all the coins should be different. If there are several solutins to that problem, print any of them.


The prestigious Codeforces kindergarten consists of n kids, numbered 1 through n. Each of them are given allowance in rubles by their parents.Today, they are going to the most famous candy shop in the town. The shop sells candies in packages: for all i between 1 and m, inclusive, it sells a package containing exactly i candies. A candy costs one ruble, so a package containing x candies costs x rubles.The kids will purchase candies in turns, starting from kid 1. In a single turn, kid i will purchase one candy package. Due to the highly competitive nature of Codeforces kindergarten, during a turn, the number of candies contained in the package purchased by the kid will always be strictly greater than the number of candies contained in the package purchased by the kid in the preceding turn (an exception is in the first turn: the first kid may purchase any package). Then, the turn proceeds to kid i + 1, or to kid 1 if it was kid n's turn. This process can be ended at any time, but at the end of the purchase process, all the kids must have the same number of candy packages. Of course, the amount spent by each kid on the candies cannot exceed their allowance.You work at the candy shop and would like to prepare the candies for the kids. Print the maximum number of candies that can be sold by the candy shop to the kids. If the kids cannot purchase any candy (due to insufficient allowance), print 0.

The first line contains two space-separated integers n and m (2 ≤ n ≤ 2·105, 2 ≤ m ≤ 5·106, n ≤ m), denoting the number of kids and the maximum number of candies in a package sold by the candy shop, respectively. Then n lines follow, each line will contain a single positive integer not exceeding  denoting the allowance of a kid in rubles. The allowances are given in order from kid 1 to kid n. Please, do not use the %lld specificator to read or write 64-bit integers in C++. It is recommended to use cin, cout streams (also you may use %I64d specificator).

Print a single integer denoting the maximum number of candies that can be sold by the candy shop.

NoteFor the first example, one of the scenarios that will result in 13 purchased candies is as follows.   Turn 1. Kid 1 purchases 1 candy.  Turn 2. Kid 2 purchases 3 candies.  Turn 3. Kid 1 purchases 4 candies.  Turn 4. Kid 2 purchases 5 candies.


After passing a test, Vasya got himself a box of $$$n$$$ candies. He decided to eat an equal amount of candies each morning until there are no more candies. However, Petya also noticed the box and decided to get some candies for himself.This means the process of eating candies is the following: in the beginning Vasya chooses a single integer $$$k$$$, same for all days. After that, in the morning he eats $$$k$$$ candies from the box (if there are less than $$$k$$$ candies in the box, he eats them all), then in the evening Petya eats $$$10\%$$$ of the candies remaining in the box. If there are still candies left in the box, the process repeats — next day Vasya eats $$$k$$$ candies again, and Petya — $$$10\%$$$ of the candies left in a box, and so on.If the amount of candies in the box is not divisible by $$$10$$$, Petya rounds the amount he takes from the box down. For example, if there were $$$97$$$ candies in the box, Petya would eat only $$$9$$$ of them. In particular, if there are less than $$$10$$$ candies in a box, Petya won't eat any at all.Your task is to find out the minimal amount of $$$k$$$ that can be chosen by Vasya so that he would eat at least half of the $$$n$$$ candies he initially got. Note that the number $$$k$$$ must be integer.

The first line contains a single integer $$$n$$$ ($$$1 \leq n \leq 10^{18}$$$) — the initial amount of candies in the box.

Output a single integer — the minimal amount of $$$k$$$ that would allow Vasya to eat at least half of candies he got.

NoteIn the sample, the amount of candies, with $$$k=3$$$, would change in the following way (Vasya eats first):$$$68 \to 65 \to 59 \to 56 \to 51 \to 48 \to 44 \to 41 \\ \to 37 \to 34 \to 31 \to 28 \to 26 \to 23 \to 21 \to 18 \to 17 \to 14 \\ \to 13 \to 10 \to 9 \to 6 \to 6 \to 3 \to 3 \to 0$$$.In total, Vasya would eat $$$39$$$ candies, while Petya — $$$29$$$.


Ivan's classes at the university have just finished, and now he wants to go to the local CFK cafe and eat some fried chicken.CFK sells chicken chunks in small and large portions. A small portion contains 3 chunks; a large one — 7 chunks. Ivan wants to eat exactly x chunks. Now he wonders whether he can buy exactly this amount of chicken.Formally, Ivan wants to know if he can choose two non-negative integers a and b in such a way that a small portions and b large ones contain exactly x chunks.Help Ivan to answer this question for several values of x!

The first line contains one integer n (1 ≤ n ≤ 100) — the number of testcases. The i-th of the following n lines contains one integer xi (1 ≤ xi ≤ 100) — the number of chicken chunks Ivan wants to eat.

Print n lines, in i-th line output YES if Ivan can buy exactly xi chunks. Otherwise, print NO.

NoteIn the first example Ivan can buy two small portions.In the second example Ivan cannot buy exactly 5 chunks, since one small portion is not enough, but two small portions or one large is too much.


Vasya and Petya wrote down all integers from 1 to n to play the "powers" game (n can be quite large; however, Vasya and Petya are not confused by this fact).Players choose numbers in turn (Vasya chooses first). If some number x is chosen at the current turn, it is forbidden to choose x or all of its other positive integer powers (that is, x2, x3, ...) at the next turns. For instance, if the number 9 is chosen at the first turn, one cannot choose 9 or 81 later, while it is still allowed to choose 3 or 27. The one who cannot make a move loses.Who wins if both Vasya and Petya play optimally?

Input contains single integer n (1 ≤ n ≤ 109).

Print the name of the winner — "Vasya" or "Petya" (without quotes).

NoteIn the first sample Vasya will choose 1 and win immediately.In the second sample no matter which number Vasya chooses during his first turn, Petya can choose the remaining number and win.


Finished her homework, Nastya decided to play computer games. Passing levels one by one, Nastya eventually faced a problem. Her mission is to leave a room, where a lot of monsters live, as quickly as possible.There are $$$n$$$ manholes in the room which are situated on one line, but, unfortunately, all the manholes are closed, and there is one stone on every manhole. There is exactly one coin under every manhole, and to win the game Nastya should pick all the coins. Initially Nastya stands near the $$$k$$$-th manhole from the left. She is thinking what to do.In one turn, Nastya can do one of the following:   if there is at least one stone on the manhole Nastya stands near, throw exactly one stone from it onto any other manhole (yes, Nastya is strong).  go to a neighboring manhole;  if there are no stones on the manhole Nastya stays near, she can open it and pick the coin from it. After it she must close the manhole immediately (it doesn't require additional moves).     The figure shows the intermediate state of the game. At the current position Nastya can throw the stone to any other manhole or move left or right to the neighboring manholes. If she were near the leftmost manhole, she could open it (since there are no stones on it). Nastya can leave the room when she picks all the coins. Monsters are everywhere, so you need to compute the minimum number of moves Nastya has to make to pick all the coins.Note one time more that Nastya can open a manhole only when there are no stones onto it.

The first and only line contains two integers $$$n$$$ and $$$k$$$, separated by space ($$$2 \leq n \leq 5000$$$, $$$1 \leq k \leq n$$$) — the number of manholes and the index of manhole from the left, near which Nastya stays initially. Initially there is exactly one stone near each of the $$$n$$$ manholes.

Print a single integer — minimum number of moves which lead Nastya to pick all the coins.

NoteLet's consider the example where $$$n = 2$$$, $$$k = 2$$$. Nastya should play as follows:  At first she throws the stone from the second manhole to the first. Now there are two stones on the first manhole.  Then she opens the second manhole and pick the coin from it.  Then she goes to the first manhole, throws two stones by two moves to the second manhole and then opens the manhole and picks the coin from it. So, $$$6$$$ moves are required to win.


Note that the memory limit in this problem is lower than in others.You have a vertical strip with $$$n$$$ cells, numbered consecutively from $$$1$$$ to $$$n$$$ from top to bottom.You also have a token that is initially placed in cell $$$n$$$. You will move the token up until it arrives at cell $$$1$$$.Let the token be in cell $$$x > 1$$$ at some moment. One shift of the token can have either of the following kinds:   Subtraction: you choose an integer $$$y$$$ between $$$1$$$ and $$$x-1$$$, inclusive, and move the token from cell $$$x$$$ to cell $$$x - y$$$.  Floored division: you choose an integer $$$z$$$ between $$$2$$$ and $$$x$$$, inclusive, and move the token from cell $$$x$$$ to cell $$$\lfloor \frac{x}{z} \rfloor$$$ ($$$x$$$ divided by $$$z$$$ rounded down). Find the number of ways to move the token from cell $$$n$$$ to cell $$$1$$$ using one or more shifts, and print it modulo $$$m$$$. Note that if there are several ways to move the token from one cell to another in one shift, all these ways are considered distinct (check example explanation for a better understanding).

The only line contains two integers $$$n$$$ and $$$m$$$ ($$$2 \le n \le 4 \cdot 10^6$$$; $$$10^8 < m < 10^9$$$; $$$m$$$ is a prime number) — the length of the strip and the modulo.

Print the number of ways to move the token from cell $$$n$$$ to cell $$$1$$$, modulo $$$m$$$.

NoteIn the first test, there are three ways to move the token from cell $$$3$$$ to cell $$$1$$$ in one shift: using subtraction of $$$y = 2$$$, or using division by $$$z = 2$$$ or $$$z = 3$$$.There are also two ways to move the token from cell $$$3$$$ to cell $$$1$$$ via cell $$$2$$$: first subtract $$$y = 1$$$, and then either subtract $$$y = 1$$$ again or divide by $$$z = 2$$$.Therefore, there are five ways in total.


$$$k$$$ people want to split $$$n$$$ candies between them. Each candy should be given to exactly one of them or be thrown away.The people are numbered from $$$1$$$ to $$$k$$$, and Arkady is the first of them. To split the candies, Arkady will choose an integer $$$x$$$ and then give the first $$$x$$$ candies to himself, the next $$$x$$$ candies to the second person, the next $$$x$$$ candies to the third person and so on in a cycle. The leftover (the remainder that is not divisible by $$$x$$$) will be thrown away.Arkady can't choose $$$x$$$ greater than $$$M$$$ as it is considered greedy. Also, he can't choose such a small $$$x$$$ that some person will receive candies more than $$$D$$$ times, as it is considered a slow splitting.Please find what is the maximum number of candies Arkady can receive by choosing some valid $$$x$$$.

The only line contains four integers $$$n$$$, $$$k$$$, $$$M$$$ and $$$D$$$ ($$$2 \le n \le 10^{18}$$$, $$$2 \le k \le n$$$, $$$1 \le M \le n$$$, $$$1 \le D \le \min{(n, 1000)}$$$, $$$M \cdot D \cdot k \ge n$$$) — the number of candies, the number of people, the maximum number of candies given to a person at once, the maximum number of times a person can receive candies.

Print a single integer — the maximum possible number of candies Arkady can give to himself. Note that it is always possible to choose some valid $$$x$$$.

NoteIn the first example Arkady should choose $$$x = 4$$$. He will give $$$4$$$ candies to himself, $$$4$$$ candies to the second person, $$$4$$$ candies to the third person, then $$$4$$$ candies to the fourth person and then again $$$4$$$ candies to himself. No person is given candies more than $$$2$$$ times, and Arkady receives $$$8$$$ candies in total.Note that if Arkady chooses $$$x = 5$$$, he will receive only $$$5$$$ candies, and if he chooses $$$x = 3$$$, he will receive only $$$3 + 3 = 6$$$ candies as well as the second person, the third and the fourth persons will receive $$$3$$$ candies, and $$$2$$$ candies will be thrown away. He can't choose $$$x = 1$$$ nor $$$x = 2$$$ because in these cases he will receive candies more than $$$2$$$ times.In the second example Arkady has to choose $$$x = 4$$$, because any smaller value leads to him receiving candies more than $$$1$$$ time.


A sequence of non-negative integers a1, a2, ..., an of length n is called a wool sequence if and only if there exists two integers l and r (1 ≤ l ≤ r ≤ n) such that . In other words each wool sequence contains a subsequence of consecutive elements with xor equal to 0.The expression  means applying the operation of a bitwise xor to numbers x and y. The given operation exists in all modern programming languages, for example, in languages C++ and Java it is marked as "^", in Pascal — as "xor".In this problem you are asked to compute the number of sequences made of n integers from 0 to 2m - 1 that are not a wool sequence. You should print this number modulo 1000000009 (109 + 9).

The only line of input contains two space-separated integers n and m (1 ≤ n, m ≤ 105).

Print the required number of sequences modulo 1000000009 (109 + 9) on the only line of output.

NoteSequences of length 3 made of integers 0, 1, 2 and 3 that are not a wool sequence are (1, 3, 1), (1, 2, 1), (2, 1, 2), (2, 3, 2), (3, 1, 3) and (3, 2, 3).


Nothing is eternal in the world, Kostya understood it on the 7-th of January when he saw partially dead four-color garland.Now he has a goal to replace dead light bulbs, however he doesn't know how many light bulbs for each color are required. It is guaranteed that for each of four colors at least one light is working.It is known that the garland contains light bulbs of four colors: red, blue, yellow and green. The garland is made as follows: if you take any four consecutive light bulbs then there will not be light bulbs with the same color among them. For example, the garland can look like "RYBGRYBGRY", "YBGRYBGRYBG", "BGRYB", but can not look like "BGRYG", "YBGRYBYGR" or "BGYBGY". Letters denote colors: 'R' — red, 'B' — blue, 'Y' — yellow, 'G' — green.Using the information that for each color at least one light bulb still works count the number of dead light bulbs of each four colors.

The first and the only line contains the string s (4 ≤ |s| ≤ 100), which describes the garland, the i-th symbol of which describes the color of the i-th light bulb in the order from the beginning of garland:    'R' — the light bulb is red,  'B' — the light bulb is blue,  'Y' — the light bulb is yellow,  'G' — the light bulb is green,  '!' — the light bulb is dead.  The string s can not contain other symbols except those five which were described.  It is guaranteed that in the given string at least once there is each of four letters 'R', 'B', 'Y' and 'G'.  It is guaranteed that the string s is correct garland with some blown light bulbs, it means that for example the line "GRBY!!!B" can not be in the input data.

In the only line print four integers kr, kb, ky, kg — the number of dead light bulbs of red, blue, yellow and green colors accordingly.

NoteIn the first example there are no dead light bulbs.In the second example it is obvious that one blue bulb is blown, because it could not be light bulbs of other colors on its place according to the statements.


Furik loves writing all sorts of problems, especially such that he can't solve himself. You've got one of his problems, the one Furik gave to Rubik. And Rubik asks you to solve it.There is integer n and array a, consisting of ten integers, indexed by numbers from 0 to 9. Your task is to count the number of positive integers with the following properties:  the number's length does not exceed n;  the number doesn't have leading zeroes;  digit i (0 ≤ i ≤ 9) occurs in the number at least a[i] times.

The first line contains integer n (1 ≤ n ≤ 100). The next line contains 10 integers a[0], a[1], ..., a[9] (0 ≤ a[i] ≤ 100) — elements of array a. The numbers are separated by spaces.

On a single line print the remainder of dividing the answer to the problem by 1000000007 (109 + 7).

NoteIn the first sample number 9 meets the requirements.In the second sample number 10 meets the requirements.In the third sample numbers 10, 110, 210, 120, 103 meet the requirements. There are other suitable numbers, 36 in total.


There is a chess tournament in All-Right-City. n players were invited to take part in the competition. The tournament is held by the following rules:  Initially, each player plays one game with every other player. There are no ties;  After that, the organizers build a complete directed graph with players as vertices. For every pair of players there is exactly one directed edge between them: the winner of their game is the startpoint of this edge and the loser is the endpoint;  After that, the organizers build a condensation of this graph. The condensation of this graph is an acyclic complete graph, therefore it has the only Hamiltonian path which consists of strongly connected components of initial graph A1 → A2 → ... → Ak.  The players from the first component A1 are placed on the first  places, the players from the component A2 are placed on the next  places, and so on.  To determine exact place of each player in a strongly connected component, all the procedures from 1 to 5 are repeated recursively inside each component, i.e. for every i = 1, 2, ..., k players from the component Ai play games with each other again, and so on;  If a component consists of a single player, then he has no more rivals, his place is already determined and the process stops. The players are enumerated with integers from 1 to n. The enumeration was made using results of a previous tournament. It is known that player i wins player j (i < j) with probability p.You need to help to organize the tournament. Find the expected value of total number of games played by all the players. It can be shown that the answer can be represented as , where P and Q are coprime integers and . Print the value of P·Q - 1 modulo 998244353.If you are not familiar with any of the terms above, you can read about them here.

The first line of input contains a single integer n (2 ≤ n ≤ 2000) — the number of players. The second line contains two integers a and b (1 ≤ a < b ≤ 100) — the numerator and the denominator of fraction .

In the only line print the expected value of total number of games played by all the players. Print the answer using the format above.

NoteIn the first example the expected value is 4.In the second example the expected value is .In the third example the expected value is .


The gym leaders were fascinated by the evolutions which took place at Felicity camp. So, they were curious to know about the secret behind evolving Pokemon. The organizers of the camp gave the gym leaders a PokeBlock, a sequence of n ingredients. Each ingredient can be of type 0 or 1. Now the organizers told the gym leaders that to evolve a Pokemon of type k (k ≥ 2), they need to make a valid set of k cuts on the PokeBlock to get smaller blocks.Suppose the given PokeBlock sequence is b0b1b2... bn - 1. You have a choice of making cuts at n + 1 places, i.e., Before b0, between b0 and b1, between b1 and b2, ..., between bn - 2 and bn - 1, and after bn - 1.The n + 1 choices of making cuts are as follows (where a | denotes a possible cut):| b0 | b1 | b2 | ... | bn - 2 | bn - 1 |Consider a sequence of k cuts. Now each pair of consecutive cuts will contain a binary string between them, formed from the ingredient types. The ingredients before the first cut and after the last cut are wasted, which is to say they are not considered. So there will be exactly k - 1 such binary substrings. Every substring can be read as a binary number. Let m be the maximum number out of the obtained numbers. If all the obtained numbers are positive and the set of the obtained numbers contains all integers from 1 to m, then this set of cuts is said to be a valid set of cuts.For example, suppose the given PokeBlock sequence is 101101001110 and we made 5 cuts in the following way:10 | 11 | 010 | 01 | 1 | 10So the 4 binary substrings obtained are: 11, 010, 01 and 1, which correspond to the numbers 3, 2, 1 and 1 respectively. Here m = 3, as it is the maximum value among the obtained numbers. And all the obtained numbers are positive and we have obtained all integers from 1 to m. Hence this set of cuts is a valid set of 5 cuts.A Pokemon of type k will evolve only if the PokeBlock is cut using a valid set of k cuts. There can be many valid sets of the same size. Two valid sets of k cuts are considered different if there is a cut in one set which is not there in the other set.Let f(k) denote the number of valid sets of k cuts. Find the value of . Since the value of s can be very large, output s modulo 109 + 7.

The input consists of two lines. The first line consists an integer n (1 ≤ n ≤ 75) — the length of the PokeBlock. The next line contains the PokeBlock, a binary string of length n.

Output a single integer, containing the answer to the problem, i.e., the value of s modulo 109 + 7.

NoteIn the first sample, the sets of valid cuts are:Size 2: |1|011, 1|01|1, 10|1|1, 101|1|.Size 3: |1|01|1, |10|1|1, 10|1|1|, 1|01|1|.Size 4: |10|1|1|, |1|01|1|.Hence, f(2) = 4, f(3) = 4 and f(4) = 2. So, the value of s = 10.In the second sample, the set of valid cuts is:Size 2: |1|0.Hence, f(2) = 1 and f(3) = 0. So, the value of s = 1.


Рассмотрим следующий код сортировки слиянием на языке Python: def sort(a):  n = len(a)  b = [0 for i in range(n)]  log = []  def mergeSort(l, r):    if r - l <= 1:      return    m = (l + r) >> 1    mergeSort(l, m)    mergeSort(m, r)    i, j, k = l, m, l    while i < m and j < r:      if a[i] < a[j]:        log.append('0')        b[k] = a[i]        i += 1      else:        log.append('1')        b[k] = a[j]        j += 1      k += 1    while i < m:      b[k] = a[i]      i += 1      k += 1    while j < r:      b[k] = a[j]      j += 1      k += 1    for p in range(l, r):      a[p] = b[p]  mergeSort(0, n)  return "".join(log)Как можно заметить, этот код использует логирование — важнейший инструмент разработки.Старший разработчик ВКонтакте Вася сгенерировал перестановку $$$a$$$ (массив из $$$n$$$ различных целых чисел от $$$1$$$ до $$$n$$$), дал её на вход функции sort и получил на выходе строку $$$s$$$. На следующий день строку $$$s$$$ Вася нашёл, а перестановка $$$a$$$ потерялась. Вася хочет восстановить любую перестановку $$$a$$$ такую, что вызов функции sort от неё даст ту же строку $$$s$$$. Помогите ему!

Ввод содержит непустую строку $$$s$$$, состоящую из символов 0 и 1.  В этой версии задачи для любого теста существует перестановка длины $$$16$$$, удовлетворяющая условию. Тем не менее, ваш ответ может иметь любую длину, в том числе отличную от $$$16$$$.

В первой строке выведите целое число $$$n$$$ — длину перестановки. Во второй строке выведите $$$n$$$ различных целых чисел $$$a_0, a_1, \ldots, a_{n-1}$$$ ($$$1 \le a_i \le n$$$) — элементы перестановки. Если существует несколько вариантов ответа, выведите любой из них.


Little Dima misbehaved during a math lesson a lot and the nasty teacher Mr. Pickles gave him the following problem as a punishment. Find all integer solutions x (0 < x < 109) of the equation:x = b·s(x)a + c,  where a, b, c are some predetermined constant values and function s(x) determines the sum of all digits in the decimal representation of number x.The teacher gives this problem to Dima for each lesson. He changes only the parameters of the equation: a, b, c. Dima got sick of getting bad marks and he asks you to help him solve this challenging problem.

The first line contains three space-separated integers: a, b, c (1 ≤ a ≤ 5; 1 ≤ b ≤ 10000;  - 10000 ≤ c ≤ 10000).

Print integer n — the number of the solutions that you've found. Next print n integers in the increasing order — the solutions of the given equation. Print only integer solutions that are larger than zero and strictly less than 109.