There is a special abacus lying flat on the table with a total of $$$m$$$ vertical rods. Each vertical rod has some beads, forming a total of $$$n$$$ rows. The $$$i$$$-th row from the top contains $$$a_i$$$ beads, where $$$a_i$$$ is a positive integer not exceeding $$$m$$$. Adjacent rows on a vertical rod may not necessarily both have beads. Now, the abacus is stood upright, and all beads fall naturally. Determine the number of beads in each row after the beads have fallen. It can be proved that there are still $$$n$$$ rows after the beads fall.

The process for the first sample case is as follows:

Computer Networks is a very important course in computer science. Suppose there are two machines, A and B, that need to transmit data through a chain-like network formed by $$$n-1$$$ routers. Specifically, when machine A wants to send data to machine B, it first sends the data to router 1, which then forwards it to router 2, router 2 forwards it to router 3, and so on, until router $$$n-1$$$ finally forwards the data to machine B.

Now, machine A needs to send $$$K$$$ bits of data to machine B. It first divides the data into $$$m$$$ groups, where the size of each group is $$$l_i$$$, satisfying $$$\Sigma l_i=K$$$. These groups are pushed to router 1 in the order of $$$l_1,l_2,...,l_m$$$. The push rate of machine A is $$$r_0$$$ bit/s, the push rate of router 1 is $$$r_1$$$bit/s, ..., and the push rate of router $$$n-1$$$ is $$$r_{n-1}$$$bit/s. The routers forward data in units of groups. When a router receives bits from a group, it waits until all bits of that group have arrived before starting to push the group to the next router. The forwarding strategy of the routers is first-come-first-served. While a router is forwarding data from one group, data from other groups is stored in the router's buffer, waiting to be forwarded. Assume that once a router or machine pushes a bit, the bit immediately arrives at the next router or machine. The question is: starting from the time when machine A pushes the first bit, at what time will machine B completely receive all $$$K$$$ bits of data sent by machine A?

There are $$$n$$$ circles on a plane. The $$$i$$$-th circle has a center at $$$(x_i,y_i)$$$ and a radius of $$$r_i$$$. These $$$n$$$ circles are pairwise non-intersecting, but they may be tangent (including internally or externally). There are two points $$$S$$$ and $$$T$$$ located on the arcs of these circles. Student A starts at point S and can only travel along the circular arcs. Each circle has a parameter $$$v_i$$$, which represents the time per unit length of traveling along the arc of that circle. The goal is to find the minimum total time for Student A to travel from point $$$S$$$ to point $$$T$$$ along the circular arcs.

Dr.Z wants to live a perfect life. In the future, he acquires a time machine that gives him a chance to alter his life. His life is represented by a string $$$S$$$. He can modify the string $$$S$$$ using another string $$$T$$$.

Each modification allows him to choose a positive integer $$$i$$$ in the range $$$[1, |S| - |T| + 1]$$$ and replace the substring $$$S[i : i + |T| - 1]$$$ with $$$T$$$. More precisely, for each $$$j \in [1, |T|]$$$, he sets $$$S_{i+j-1} = T_j$$$. This operation can be performed infinitely many times.Dr.Z considers his life perfect if and only if $$$\forall i \in [1, |S|], S_i = S_{|S| - i + 1}$$$—that is, when the string $$$S$$$ becomes a palindrome.

Dr.Z wants to know whether it's possible for his life to become perfect through these modifications.

There are $$$t$$$ test cases. For each test case, you are given two strings $$$S$$$ and $$$T$$$.If $$$S$$$ can become a palindrome, output Yes. Otherwise, output No.

Dr.Orange is conducting research in animal psychology, and one of the studies involves inferring the number of each animal in the ecosystem based on their dietary habits. It is known that there are $$$n$$$ types of animals in the ecosystem, and the $$$i$$$-th type of animal consumes $$$a_i$$$ portions of bait per time. Dr.Orange's experiment was conducted over $$$m$$$ days, and on the $$$i$$$-th day, $$$t_i$$$ portions of bait were deployed, with all bait being consumed each day. For each of the $$$m$$$ days, different animals take turns to consume bait. You need to answer how many different eating sequences. Two sequences are the same only if the animals at each position in the sequence are of the same species. Since this result can be very large, please provide the answer modulo $$$10^9+7$$$.

As you all know, you are participating in the SYSUCPC (SYSU Counting Periodic-string Contest) finals.

A string is considered good if and only if it can be rearranged into a perfectly periodic string with sysu as its period (which means several sysu repetitions). For example, sysu,yssu,ssyyssuu are all good, while ussu, sysusys are not. Given a string $$$S$$$, determine how many of its substrings are good.

A string $$$a$$$ is a substring of a string $$$b$$$ if $$$a$$$ can be obtained from $$$b$$$ by the deletion of several (possibly zero or all) characters from the beginning and several (possibly zero or all) characters from the end.

Gray code is a binary numeral system where two successive values differ in only one bit. This property makes it useful in various applications, such as minimizing errors in digital communication and measurement systems. Gray code is also known for its reflective and cyclic characteristics, which help eliminate significant errors during random sampling. It was invented by Frank Gray in the 1940s and has various construction methods.

An $$$n$$$-bit Gray code $$$G_n=[G_{n,0},G_{n,1},\cdots,G_{n,2^n-1}]$$$ can be generated by the following algorithm:

1. The $$$1$$$-bit Gray code consists of the two binary numbers $$$0$$$ and $$$1$$$. That is, $$$G_1=[0,1]$$$.
2. The first $$$2^n$$$ binary numbers of the $$$(n+1)$$$-bit Gray code are the same as the $$$n$$$-bit Gray code; the last $$$2^n$$$ binary strings of the $$$(n+1)$$$-bit Gray code are formed by taking the $$$n$$$-bit Gray code, arranging it in reverse order, and then adding $$$2^n$$$ to each number. That is, $$$G_{n+1}=[G_{n,0},G_{n,1},\cdots,G_{n,2^n-1},G_{n,2^n-1}+2^n,G_{n,2^n-2}+2^n,\cdots,G_{n,0}+2^n]$$$.

For example, $$$G_2=[0,1,3,2]$$$, $$$G_3=[0,1,3,2,6,7,5,4]$$$.

Now there is an array $$$a_0,a_1,\cdots,a_{2^n-1}$$$ of length $$$2^n$$$. Initially, for all $$$i\in\{0,1,\cdots,2^n-1\}$$$, $$$a_i=i$$$. We will perform $$$q$$$ operations on this array, where each operation is one of the following two types:

1. Given $$$m$$$ and $$$k_1,k_2,\cdots,k_m$$$, for $$$i=1,2,\cdots,m$$$ in increasing order, for every $$$j\in\{0,1,\cdots,2^{k_i}-1\}$$$, if $$$a_j \lt 2^{k_i}$$$, then set $$$a_j=G_{k_i,a_j}$$$; otherwise, do nothing to $$$a_j$$$.
2. Given $$$p$$$, query the value of $$$a_p$$$.

Please output the results of each query in order.

As you all know, you are participating in the SYSUCPC (SYSU Constructive Problem Contest) finals. Undoubtedly, the Constructive Kingdom of Sun Yat-sen University is eager for you to tackle some related problems.

Given a string $$$S$$$ containing only lowercase letters, determine the maximum number of disjoint subsequences of length $$$4$$$ that can be formed, such that all subsequences are sysu. Formally, a subsequence of a string $$$S$$$ is a subset of characters of $$$S$$$ that read from left to right.

A greedy algorithm is currently used: each time, starting from the end, an unused u is considered, and it is matched with nearest unused s,y,s sequentially to form a qualified subsequence, which is then removed.

You are now requested to provide a string $$$S$$$ such that the answer obtained by the greedy method is $$$x$$$, while the correct answer is $$$y$$$. If no such $$$S$$$ exists, output $$$-1$$$.

Student A is studying numeral systems. For a natural number $$$n$$$, Student A wants to know the smallest digit sum among the representations of $$$n$$$ in bases from $$$2$$$ to $$$R$$$. For example, the natural number $$$6$$$ has a representation of $$$(110)_2$$$ in base $$$2$$$, with a digit sum of $$$2$$$, and a representation of $$$(20)_3$$$ in base $$$3$$$, also with a digit sum of $$$2$$$.

Recently, Student A has been studying strings. Certain peculiar strings caught his attention, so he named such strings "quasi-palindromic strings".

A string $$$s$$$ is a quasi-palindromic string if and only if the following two conditions are both satisfied:

1. There does not exist any contiguous substring $$$t$$$ of $$$s$$$ with length greater than or equal to 2 that is a palindromic string.

2. Suppose the length of $$$s$$$ is $$$n$$$, there exist $$$l,r$$$ satisfying $$$1\leq l\leq r\leq n$$$, such that after reversing the substring $$$s[l...r]$$$, $$$s$$$ becomes a palindromic string.

For example: The string "aac" is not a quasi-palindromic string because it contains the contiguous substring "aa" of length 2, which is a palindrome. The string "abdc" is not a quasi-palindromic string because there do not exist $$$l,r$$$ such that reversing the substring $$$s[l...r]$$$ would turn $$$s$$$ into a palindrome. The string "bcabca" is a quasi-palindromic string.

Now, given a string $$$T$$$, Student A wants to know how many contiguous substrings of $$$T$$$ are quasi-palindromic strings.

Here, contiguous substrings are distinguished solely by their starting and ending positions in the original string. For instance, in the string "aba", the contiguous substring [1,1] (the first "a") and the contiguous substring [3,3] (the last "a") are considered two different contiguous substrings, even though they are both the character "a".

Dr.Z is studying a permutation $$$p$$$ of length $$$n$$$ (i.e., $$$p$$$ contains each integer from $$$1$$$ to $$$n$$$ exactly once). He performs $$$q$$$ experiments, each defined by parameters $$$(l, r, x)$$$.

In each experiment, he processes the subarray $$$p_l, p_{l+1}, \dots, p_r$$$ sequentially. Starting with value $$$x$$$, for each element $$$p_i$$$ in the subarray:

• If $$$x \mid p_i$$$ ($$$x$$$ divides $$$p_i$$$), then $$$x$$$ becomes $$$p_i / x$$$
• Else if $$$p_i \mid x$$$ ($$$p_i$$$ divides $$$x$$$), then $$$x$$$ becomes $$$x / p_i$$$
• Otherwise, $$$x$$$ remains unchanged

Dr.Z needs your help to compute:

1. The final value of $$$x$$$ after processing each experiment
2. The total number of times either $$$x \mid p_i$$$ or $$$p_i \mid x$$$ was satisfied during the process

In his recent work on permutation properties, Dr.Z investigates the relationship between element values and their indices. A key focus is the cardinality of the set of indices $$$i$$$ where $$$p_i \gt i$$$ (called larger position) and where $$$p_i \lt i$$$ (called smaller position). To formalize this, he defines $$$f(n,x,y)$$$ as the number of permutations of $$$n$$$ elements containing exactly $$$x$$$ larger positions and $$$y$$$ smaller positions. Dr.Z now aims to characterize this function completely by determining the values of $$$f(n,x,y)$$$ across all valid positive integers $$$x$$$ and $$$y$$$ after modulo $$$m$$$. Could you help him solve this problem efficiently?

The highway system of Province G can be viewed as a weighted undirected graph with $$$n$$$ nodes and $$$m$$$ edges. Each edge represents a segment of the highway, and each node represents a city. Every highway segment is bidirectional. The $$$i$$$-th highway segment connects the two cities $$$u_i$$$ and $$$v_i$$$, with a length of $$$w_i$$$ kilometers.

Miss Orange is currently commuting in Province G. Her car consumes one unit of fuel per kilometer traveled. Any city has a gas station where the car's fuel tank can be fully refilled, and the fuel consumption of Miss Orange's car while driving within the city is negligible. Miss Orange has planned $$$q$$$ trips. The $$$i$$$-th trip starts from city $$$x$$$ and ends at city $$$y$$$. She wants to know the minimum fuel tank capacity required for this trip.

However, she has not decided on $$$x$$$ and $$$y$$$ but only knows that $$$x$$$ and $$$y$$$ lie within the interval $$$[l_i, r_i]$$$. Specifically, for all $$$l_i \le x \lt y \le r_i$$$, Miss Orange wants to know the sum of the minimum fuel tank capacities required to travel from $$$x$$$ to $$$y$$$. Formally, let $$$f(x, y)$$$ denote the fuel tank capacity required for a trip starting at $$$x$$$ and ending at $$$y$$$. You need to output $$$\sum\limits_{l_i \le x \lt y \le r_i} f(x, y)$$$.

Since she needs to consider her next plan based on the current one, certain constraints will be imposed on you in the problem.