Little A has a rectangular box with a height of $$$h$$$. He places this box on a table and establishes a three-dimensional Cartesian coordinate system $$$O-xyz$$$, where the table is located at $$$z=z_0$$$.

Due to Little A's obsessive-compulsive disorder, he places the length and width of the box parallel to the $$$x$$$ and $$$y$$$ axes (the length does not necessarily correspond to the $$$x$$$ axis), and the bottom face has a pair of diagonal vertices located at $$$(u_0,v_0,z_0)$$$ and $$$(u_1,v_1,z_0)$$$.

Now Little A has $$$q$$$ queries. For each query, he provides a point $$$(x_i,y_i,z_i)$$$ and wants to know if the point is inside the box (the boundary is also considered inside).

Little A has many strings, each with a character set of $$$\{1,2,\cdots,m\}$$$. He constructed a Trie for these strings and built an Aho-Corasick automaton from this Trie.

However, due to Little A's negligence, both the original strings and the constructed Aho-Corasick automaton have disappeared. Little A only remembers that the lengths of all original strings do not exceed $$$d$$$, and that the constructed Aho-Corasick automaton has $$$n$$$ vertices and a character set of $$$\{1,2,\cdots,m\}$$$.

Now, Little A wants to know how many different Aho-Corasick automatons could possibly be the one he constructed. Since the answer can be large, you only need to find the result modulo $$$998244353$$$.

The Aho-Corasick automaton is defined as follows:

1. A Trie $$$T$$$ is a rooted tree without labels, where each edge is labeled with a character. A vertex $$$x$$$ in the Trie cannot have two child vertices $$$y$$$ and $$$z$$$ such that the edges $$$(x, y)$$$ and $$$(x, z)$$$ are labeled with the same character.
2. Given a Trie $$$T$$$ with root $$$r$$$, for a vertex $$$x$$$, the string represented by $$$x$$$ is the concatenation of the characters on the edges from $$$r$$$ to $$$x$$$. In particular, the string represented by $$$r$$$ is the empty string. It can be proven that no two different vertices represent the same string.
3. We say that a string $$$S$$$ exists in Trie $$$T$$$ if and only if there exists a vertex $$$x$$$ such that the string represented by $$$x$$$ is $$$S$$$.
4. Constructing a Trie $$$T$$$ for some strings $$$S_1,S_2,\cdots,S_k$$$ means finding a Trie $$$T$$$ with the minimum number of vertices such that all strings $$$S_i$$$ exist in Trie $$$T$$$. It can be proven that such a Trie is unique when unlabeled.
5. The fail tree $$$F$$$ of Trie $$$T$$$ is a tree with root $$$r$$$. Define $$$S_x$$$ as the string represented by vertex $$$x$$$. For a non-root vertex $$$x$$$, let $$$U$$$ be the longest proper suffix of $$$S_x$$$ (a proper suffix is one that is not equal to $$$S_x$$$) such that $$$U$$$ exists in $$$T$$$. Then, $$$fail_x$$$ is defined as the vertex such that $$$S_{fail_x} = U$$$. Note that the empty suffix of $$$S_x$$$ always exists in $$$T$$$, so $$$fail_x$$$ always exists. The edge set of $$$F$$$ is $$$\{(x, fail_x) | x \in [1, n], x \neq r\}$$$. It can be proven that these edges form a tree.
6. Merging Trie $$$T$$$ and its fail tree $$$F$$$ means that the vertex set remains unchanged (the vertex sets of $$$T$$$ and $$$F$$$ are the same), and the edges and the characters on the edges are merged to obtain the graph, which is the Aho-Corasick automaton $$$A$$$ of Trie $$$T$$$. At this point, $$$A$$$ contains both edges labeled with characters and edges without labels.

Two Aho-Corasick automatons are considered the same if and only if they have the same number of vertices and there exist two labeling schemes for the vertices (let's assume they are two permutations of length equal to the number of vertices) such that:

1. The roots of the two Aho-Corasick automatons are the same.
2. For any pair of vertices $$$x,y$$$, either there is no edge between $$$x$$$ and $$$y$$$ in both Aho-Corasick automatons, or there is an edge in both and the characters on the edges are the same, or both edges do not have labels.

Given a string $$$S$$$ composed of uppercase letters, please rearrange the order of the characters in $$$S$$$ such that the substring CCPC appears as many times as possible as a contiguous substring. You are to determine the maximum possible number of occurrences of CCPC.

Little H has a permutation $$$P$$$, and he wants to split $$$P$$$ into sequence $$$A$$$ and sequence $$$B$$$.

Specifically, Little H will take several elements from $$$P$$$ in order and place them into sequence $$$A$$$, while the remaining elements will form another sequence $$$B$$$ in order.

For example, if $$$P = [1,2,3,4,5]$$$, he can split $$$P$$$ into $$$A = [1,3,5], B = [2,4]$$$.

He is very fond of increasing subsequences and decreasing subsequences. Define $$$f(A)$$$ as the length of the longest increasing subsequence of $$$A$$$, and $$$g(B)$$$ as the length of the longest decreasing subsequence of $$$B$$$. You need to tell him the maximum value of $$$f(A) + g(B)$$$.

Given a rooted tree with $$$n$$$ nodes, rooted at $$$1$$$, which satisfies the property $$$p_i \lt i$$$, where $$$p_i$$$ is the parent node of $$$i$$$.

For each node $$$u$$$, for all its children $$$v$$$, we will provide the index of the centroid of the new tree formed by only considering $$$v$$$ and the nodes in the subtree of $$$v$$$, noting that we will not give you the index of $$$v$$$. If a tree has two centroids, we will tell you the one that is deeper in the original tree. Your task is to construct a tree that satisfies the above conditions.

The centroid of a tree: If a certain node is chosen and removed from the tree, the tree will be divided into several subtrees. The number of nodes in each subtree is counted, and the maximum value is recorded. The node for which this maximum value is minimized when considering all nodes in the tree is called the centroid of the entire tree.

Little A has a sequence of positive integers of length $$$n$$$, denoted as $$$a_1, a_2, \cdots, a_n$$$. He wishes to create another sequence of positive integers of length $$$n$$$, denoted as $$$d_1, d_2, \cdots, d_n$$$, such that $$$d_i$$$ is a divisor of $$$a_i$$$.

It is evident that $$$d_1, d_2, \cdots, d_n$$$ are not unique, so Little A hopes that the product of $$$d_1, d_2, \cdots, d_n$$$ is a perfect square $$$x = y^2$$$, where $$$y$$$ is a positive integer.

However, at this point, $$$d_1, d_2, \cdots, d_n$$$ are still not unique. Therefore, Little A wants to know the sum of $$$y$$$, the square root of the product for all possible combinations of $$$d_1, d_2, \cdots, d_n$$$ that yield a perfect square $$$x = y^2$$$, modulo $$$10^9 + 7$$$.

Given a non-negative integer sequence $$$a$$$ of length $$$n$$$ and a constant value $$$k$$$.

Determine how many integers $$$x$$$ satisfy $$$x \in [0,k]$$$, such that $$$a_1 \oplus x, a_2 \oplus x, \ldots, a_n \oplus x$$$ forms a non-decreasing sequence.

Here, $$$\oplus$$$ denotes the XOR operation.

Little A has a string that only contains the characters 0 and 1 (hereafter referred to as a 01 string). He likes 1 and dislikes 0, so in Little A's eyes, there are only 1s in a 01 string.

Specifically, for a 01 string, Little A sees 0 as a separator that divides the string into several substrings consisting entirely of 1s. For example, for a 01 string 010011101111101, Little A sees four substrings that consist only of 1s: 1, 111, 11111, and 1.

For a 01 string, Little A defines its charm value as the sum of the square roots of the lengths of the separated substrings that consist entirely of 1s. For example, for the string 010011101111101, Little A's charm value is $$$\sqrt{1}+\sqrt{3}+\sqrt{5}+\sqrt{1}=2+\sqrt{3}+\sqrt{5}$$$.

Now, given a 01 string $$$s$$$, Little A hopes you can change some of the 1s in $$$s$$$ to 0s (or leave them unchanged) to maximize the charm value of this 01 string.

Little H initially has a string $$$s$$$ composed of lowercase letters.

The charm value of a string is defined as the number of essentially different substrings.

For example, $$$\texttt{aaa}$$$ has only $$$3$$$ essentially different substrings: $$$\texttt{a}, \texttt{aa}, \texttt{aaa}$$$, while $$$\texttt{aabb}$$$ has $$$8$$$ essentially different substrings: $$$\texttt{a}, \texttt{aa}, \texttt{b}, \texttt{bb}, \texttt{ab}, \texttt{aab}, \texttt{abb}, \texttt{aabb}$$$.

He thinks the charm value of the initial string $$$s$$$ is too low, so he duplicates $$$s$$$ $$$m$$$ times and concatenates them together, trying to obtain a string with a higher charm value.

However, after he finished duplicating, he found that he could not accurately calculate its charm value. Please help him calculate the charm value of the duplicated string. Since the answer may be large, you need to output the charm value modulo $$$998244353$$$.

Little H has two permutations $$$a$$$ and $$$b$$$, both of length $$$n$$$.

There are $$$q$$$ queries, and for each query, he will give you an interval $$$[l,r]$$$ for $$$a$$$ and an interval $$$[L, R]$$$ for $$$b$$$. He wants to know the value of $$$\sum\limits_{i=l}^r \sum\limits_{j=L}^R \gcd^2(a_i,b_j)$$$, but you only need to provide the result modulo $$$2^{32}$$$.

In the legendary magical land, there is a mysterious school. Xiao Kai is a student at this school, and recently his class is evaluating for the Excellent Student Scholarship and the Provincial Government Scholarship.

First, for each class, the allocation of Excellent Student Scholarship slots is as follows: Suppose a class has $$$n$$$ students, then at most $$$\lfloor{0.15n}\rfloor$$$ students will receive the first prize scholarship, at most $$$\lfloor{0.25n}\rfloor$$$ students will receive the second prize scholarship, and at most $$$\lfloor{0.35n}\rfloor$$$ students will receive the third prize scholarship. (For example, in a class of 21 students, 3 students will receive the first prize scholarship, 5 students will receive the second prize scholarship, and 7 students will receive the third prize scholarship.)

Each student will have scores in three areas: academic performance, moral education, and physical education, with scores being integers in the range $$$[0,100]$$$. The comprehensive score is the sum of the three scores. When evaluating scholarships, students will first be sorted in descending order by their comprehensive scores. If two students have the same comprehensive score, they will be sorted by their academic performance scores in descending order. If both comprehensive and academic scores are the same, they will be sorted by their names in ascending lexicographical order. The resulting ranking is referred to as the comprehensive ranking. The college will determine the order of scholarship applications based on the comprehensive ranking.

In addition, there is a rule that restricts the level of scholarship that students can apply for: students receiving the first prize scholarship must ensure that their academic performance score is in the top $$$25\%$$$ of the class, students receiving the second prize scholarship must ensure that their academic performance score is in the top $$$45\%$$$, and students receiving the third prize scholarship must ensure that their academic performance score is in the top $$$75\%$$$. (For example, in a class of 21 students, only the top 5 students in academic performance are eligible to receive the first prize scholarship, the top 9 students are eligible for the second prize scholarship, and the top 15 students are eligible for the third prize scholarship. Specially, if two students are tied for fifth place in academic performance, both are eligible to receive the first prize scholarship.)

After the Excellent Student Scholarships for the two semesters are awarded, the college will also evaluate the Provincial Government Scholarship. The college stipulates that there are only $$$m$$$ slots for the Provincial Government Scholarship in Xiao Kai's class. The evaluation method is to first sort by award points (where one first prize scholarship counts as 15 points, one second prize scholarship counts as 10 points, and one third prize scholarship counts as 5 points; for example, if student X receives one first prize scholarship and one second prize scholarship, their award points will be 25 points), then if the award points are the same, sort by the total comprehensive score in descending order, if the total comprehensive scores are the same, sort by the total academic performance score from both semesters in descending order, and if both total comprehensive scores and total academic performance scores are the same, sort by names in ascending lexicographical order.

Unfortunately, Xiao Kai was unable to win the Provincial Government Scholarship, which made him feel down. That night, he had a dream where he met an immortal sitting on a gourd, who could help him fulfill his wish of winning the Provincial Government Scholarship in his dream. The immortal can sell Xiao Kai several cups of drinks, where the first type of drink costs $$$p$$$ gold coins and can increase Xiao Kai's academic performance score in the first semester by 1 point, and the second type of drink costs $$$q$$$ gold coins and can increase Xiao Kai's academic performance score in the second semester by 1 point. (Note that each semester's academic performance score has a maximum limit of 100 points.)

Given that Xiao Kai's name is registered as crazyzhk, he now tells you all the scores of his class for the two semesters, and he wants to ask you how many gold coins he needs at a minimum to drink the purchased beverages and be able to win the Provincial Government Scholarship. If Xiao Kai cannot win the Provincial Government Scholarship no matter what, please output "Surely next time" (without quotes) to encourage him.

What is lexicographical order:

In simple terms, lexicographical order means "the order in which words appear in a dictionary." More accurately, the algorithm to determine the order of two different strings $$$S$$$ and $$$T$$$ composed of lowercase letters is as follows:

Let the $$$i$$$-th character of $$$S$$$ be denoted as $$$S_i$$$.

Define that if $$$S$$$ is lexicographically less than $$$T$$$, we consider $$$S \lt T$$$, and if $$$S$$$ is lexicographically greater than $$$T$$$, we consider $$$S \gt T$$$.

• Let $$$L$$$ be the length of the shorter string between $$$S$$$ and $$$T$$$. We check $$$S_i$$$ and $$$T_i$$$ for $$$i=1,2,\cdots,L$$$ in order.
• If there exists an $$$i$$$ such that $$$S_i \neq T_i$$$, let $$$j$$$ be the smallest $$$i$$$ that satisfies this condition. Compare $$$S_j$$$ and $$$T_j$$$. If $$$S_j$$$ is lexicographically less than $$$T_j$$$, then $$$S \lt T$$$. Otherwise, $$$S \gt T$$$. The algorithm ends here.
• If there is no $$$i$$$ such that $$$S_i \neq T_i$$$, then we compare the lengths of $$$S$$$ and $$$T$$$. If the length of $$$S$$$ is less than that of $$$T$$$, then $$$S \lt T$$$. If the length of $$$S$$$ is greater than that of $$$T$$$, then $$$S \gt T$$$. If the lengths of $$$S$$$ and $$$T$$$ are equal, then $$$S=T$$$. The algorithm ends here.

$$$2\times 2$$$ and $$$3\times 3$$$ rectangle examples, A, B, C, D represent four basic blocks

Little Y wants to play a type of puzzle that consists of four basic blocks, as shown in the image above. Little Y has $$$A, B, C, D$$$ of these four types of blocks. Now he wants to use as many blocks as possible to form a rectangle (either a rectangle or a square). The question is, what is the maximum number of blocks he can use?

The formed rectangle must satisfy the following conditions:

1. Any two adjacent basic blocks must form a concave and convex structure;
2. The four edges of the formed rectangle must not have any protrusions or indentations;
3. Unlike common puzzle games, the blocks in this game do not have patterns, and blocks of the same type are considered identical.

Given a tree with $$$n$$$ nodes, you can use several line segments of arbitrary lengths to cover all the edges of the tree, with the following requirements:

1. Each edge must be covered exactly once;
2. Each line segment must start from a leaf node and end at one of its ancestor nodes.

You can choose any number of line segments such that these segments can cover the entire tree according to the above requirements, but you need to minimize the maximum length of the segments. Your task is to find this minimum value.