Given an array $$$a$$$ of length $$$n$$$, perform the following operation exactly once:

— Delete an element from the array, then concatenate the resulting parts (i.e., if you remove the $$$i$$$-th element, the array becomes $$$a_1, a_2, \dots, a_{i - 1}, a_{i + 1}, \dots, a_n$$$).

Let $$$b_1, b_2, \dots, b_{n - 1}$$$ be the resulting array after performing the above operation exactly once on the array $$$a$$$.

Find the maximum possible sum of the values of GCD of each prefix in the array $$$b$$$.

More formally, find the maximum value of: $$$$$$\sum_{i = 1}^{n - 1} gcd(b_1, b_2, \dots, b_i)$$$$$$

Over all the arrays $$$b$$$ resulting from applying the mentioned operation on the array $$$a$$$ exactly once.

Note: $$$gcd(x_1, x_2, \dots, x_k)$$$ is the greatest integer that divides all the values $$$x_1, x_2, \dots, x_k$$$ evenly without a remainder.

Given a tree consisting of $$$n$$$ nodes.

Is there a walk that visits all the nodes of the tree at least once and at most twice?

Note that the node you start the walk at is considered visited at the beginning.

A walk is a sequence of nodes $$$v_1, v_2, \dots, v_k$$$ such that for each $$$i$$$ ($$$1 \le i \lt k$$$), $$$v_i$$$ and $$$v_{i + 1}$$$ are connected by an edge.

Define the power of a string $$$t$$$ of length $$$m$$$ as follows:

$$$$$$\prod_{i = 1}^{m} i^{f_t(t_i)}$$$$$$

Where:

— $$$t_i$$$ is the character at position $$$i$$$ of the string $$$t$$$.

— $$$f_t(c)$$$ is the number of occurrences of the character $$$c$$$ in the string $$$t$$$.

For example, if $$$t = $$$ "abca", then $$$f($$$'a'$$$) = 2$$$, $$$f($$$'b'$$$) = 1$$$, and $$$f($$$'c'$$$) = 1$$$, and the power of $$$t$$$ is equal to $$$1^2 \times 2^1 \times 3^1 \times 4^2 = 96$$$.

Given a string $$$s$$$ of length $$$n$$$, and $$$q$$$ queries of the following form:

— "$$$l, r$$$": determine whether it is possible to swap two characters in $$$t[l \dots r]$$$ such that the power of $$$t[l \dots r]$$$ increases.

A grid of size $$$n \times n$$$ is constructed in the following way:

The first row of the grid consists of the numbers $$$1, 2, 3, \dots, n$$$ (in that order).

Then, each other cell in the grid will be filled with the square of the number in the cell above it.

Given an integer $$$n$$$, count the number of distinct integers in an $$$n \times n$$$ grid.

Define $$$mex(b)$$$ as the minimum non-negative integer that doesn't appear in the array $$$b$$$. For example, $$$mex([1, 0, 2]) = 3$$$ and $$$mex([1]) = 0$$$.

Given an array $$$a$$$ of length $$$n$$$. You can do the following operation at most once:

• Choose two integers $$$l$$$ and $$$r$$$ ($$$1 \le l \le r \le n$$$), and let $$$m = mex(a[l \dots r])$$$ (where $$$a[l \dots r]$$$ is $$$a_l, a_{l + 1}, \dots a_r$$$).
• For each $$$i$$$ ($$$l \le i \le r$$$), assign $$$a_i$$$ = $$$m$$$.

What is the maximum value of $$$mex(a)$$$ after performing the above operation at most once.

Given a string $$$s$$$ of length $$$n$$$ consisting of lowercase English letters.

You have to answer $$$q$$$ queries of the following form:

— "$$$l$$$ $$$r$$$": Find the minimum size of substring $$$s[i \dots j]$$$ ($$$l \le i \le j \le r$$$) such that the number of distinct characters in $$$s[i \dots j]$$$ is equal to the number of distinct characters in $$$s[l \dots r]$$$.

Given a string $$$s$$$ consisting of lowercase English letters.

You can do the following operation on $$$s$$$:

— Choose some substring $$$s[l \dots r]$$$, and let $$$x$$$ be the number of distinct characters in $$$s[l \dots r]$$$.

— Change all characters in $$$s[l \dots r]$$$ to the $$$x$$$-th character in the alphabet (e.g., the first character is 'a', the second is 'b', and so on).

You apply the above operation any number of times (possibly zero) such that $$$\textbf{no index gets changed more than once}$$$.

What is the lexicographically maximum string that could be obtained?

The city of Banis has a hotel consisting of $$$n$$$ floors.

The $$$i$$$-th floor (from the bottom) can withstand a weight of $$$w_i$$$ (i.e., there should be no more than $$$w_i$$$ clients on the $$$i$$$-th floor $$$\textbf{or above}$$$, otherwise, the hotel would collapse).

The hotel manager expects $$$m$$$ clients to come to the hotel this month, and each client has a certain rank.

If the rank of client $$$i$$$ is $$$r_i$$$ ($$$1 \le r_i \le n$$$), then the manager would want them to be on the $$$r_i$$$-th floor or lower.

However, he does not know the ranks of the clients. Therefore, $$$r_i$$$ can be from $$$1$$$ to $$$n$$$ with equal probability.

If a client is placed on the $$$i$$$-th floor, they would give the hotel a rating of $$$i$$$.

As a part of the hotel's staff, you need to place the clients optimally to get the maximum possible sum of ratings (but while still satisfying the conditions mentioned above).

Let $$$X$$$ be the sum of the ratings that each client gives.

What is the expected value of $$$X$$$ if you place the $$$m$$$ clients optimally? You should print the answer modulo $$$10 ^ 9 + 7$$$.

Given an array $$$a$$$ of length $$$n$$$, and you should answer $$$q$$$ queries of the following form:

— "$$$x$$$": Find the minimum value of $$$a_i \oplus a_j$$$ ($$$1 \le i \lt j \le n$$$) such that ($$$a_i \: \: | \: \: a_j \: \: | \: \: x \: = \: x$$$).

Note that $$$\oplus$$$ and $$$|$$$ denote the bitwise $$$XOR$$$ and the bitwise $$$OR$$$ operations, respectively.

It's hard to predict the winner when legends face each other, especially in chess.

Ahmad and Ahmad are playing a series of $$$n$$$ chess matches, and one of them will win more games than the other, as there will be no ties and $$$n$$$ is odd.

Can you predict the winner?

You are given $$$n$$$ distinct points $$$(x_1, y_1), (x_2, y_2), \dots, (x_n, y_n)$$$.

Each point $$$i$$$ has a color $$$c_i$$$.

Find the $$$\textbf{squared}$$$ Euclidean distance between the farthest pair of points that are not from the same color.

Note: The Euclidean distance between two points $$$(a_1, b_1)$$$ and $$$(a_2, b_2)$$$ is defined as follows: $$$\sqrt{(a_1 - a_2)^2 + (b_1 - b_2)^2}$$$.

Define the median of an array $$$b_1, b_2, \dots b_k$$$ as $$$b_{\lfloor \frac{k + 1}{2} \rfloor}$$$ after sorting the array $$$b$$$.

For example, the median of the array $$$b = [5, 4, 6, 1, 1]$$$ is $$$4$$$ because after sorting the array becomes $$$b = [1, 1, 4, 5, 6]$$$ and $$$b_{\lfloor \frac{5 + 1}{2} \rfloor} = b_{\lfloor \frac{6}{2} \rfloor} = b_3 = 4$$$. In the same way, the median of the array $$$[3, 5, 2, 7]$$$ is $$$3$$$.

Given an array $$$a_1, a_2, \dots, a_n$$$, is it possible to divide it into two non-empty arrays $$$b$$$ and $$$c$$$ (not necessarily of the same length), such that each element of $$$a$$$ is either in $$$b$$$ or in $$$c$$$, and the medians of both $$$b$$$ and $$$c$$$ are equal?

Note: $$$\lfloor x \rfloor$$$ denotes the integer part of $$$x$$$ (e.g., $$$\lfloor 5.3 \rfloor = 5$$$ and $$$\lfloor 2.7 \rfloor = 2$$$).

Kaito has $$$n$$$ Zingers, and he wants to distribute them between the Kaito kciD team members.

Kaito kciD team consists of 3 members, one of them is greedy Taim.

In order to distribute the $$$n$$$ Zingers between them, Kaito will follow the steps below in order:

— If $$$n$$$ is divisible by $$$3$$$, each of them will get $$$\frac{n}{3}$$$ Zingers.

— Otherwise, if $$$n$$$ is divisible by $$$2$$$, Taim will get $$$\frac{N}{2}$$$ Zingers, and the other two will get the rest.

— Otherwise, Taim will get all the $$$n$$$ Zingers.

Taim can steal at most $$$k$$$ Zingers before Kaito distributes them among the team.

What is the maximum number of Zingers Taim could have (stolen Zingers included)?