For a string $$$s$$$, let $$$f(s)$$$ be the string obtained by merging possibly several consecutive equal characters into a single character. For example, $$$f(abbbccdc) = abcdc$$$.

Given a string $$$a$$$, find the shortest substring $$$b$$$ of $$$a$$$ such that $$$f(a)$$$ is equal to $$$f(b)$$$.

Given integers $$$n$$$ and $$$k$$$, construct a permutation $$$p$$$ of integers $$$1 ... n$$$ that maximizes the value of $$$k$$$ after the following process.

Perform $$$n$$$ operations, on the $$$i$$$-th operation ($$$i = 1, 2, ..., n$$$)

• It if $$$i$$$ is odd, $$$k := p_i \cdot k$$$.
• if $$$i$$$ is even, $$$k := \lfloor \frac{k}{p_i} \rfloor$$$.

Given $$$3$$$ integers $$$x$$$, $$$y$$$, and $$$z$$$, you can perform three types of operations on them.

Swap the values of any $$$2$$$ of them ($$$x$$$ and $$$y$$$ or $$$x$$$ and $$$z$$$ or $$$y$$$ and $$$z$$$).

Choose one of them and increase the value of it by $$$1$$$

Choose one of them that is greater than $$$1$$$ and decrease the value of it by $$$1$$$.

Whats the minimum number of operations to make $$$x$$$ a factor of both $$$y$$$ and $$$z$$$?

You are given a strictly increasing array $$$a$$$ of length $$$n$$$ ($$$a_1 \lt a_2 \lt \ldots \lt a_n$$$).

Determine whether it is possible to insert $$$k$$$ integers $$$b_1 \le b_2 \le \ldots \le b_k$$$ arbitrarily into $$$a$$$, such that $$$b_1 \ge a_1$$$, $$$b_k \le a_n$$$, and $$$a$$$ is strictly increasing after all $$$k$$$ integers have been inserted.

Since this problem is too easy, solve the problem for $$$q$$$ point updates on $$$a$$$.

You are given an array $$$a$$$ of length $$$n$$$, and an initially empty array $$$b$$$.

You may perform the following operation at most once:

1. Pick two arbitrary indices $$$l$$$ and $$$r$$$ ($$$1 \le l \le r \le n$$$).
2. Replace $$$a_l, a_{l + 1}, \ldots, a_r$$$ with it's cyclic shift to the left by an offset of $$$1$$$.

For instance, choosing the indices $$$l = 2$$$ and $$$r = 5$$$ for the array $$$[1, 5, 2, 3, 4, 6]$$$ will result in the new array $$$[1, 2, 3, 4, 5, 6]$$$ after the operation is performed.

After performing the operation (if you choose to do so), you will build another array $$$b$$$ with the following process: on the $$$i$$$th turn of the process, you can insert $$$a_i$$$ at either the front or back of $$$b$$$.

Determine if there exists a way to construct $$$b$$$ such that $$$b$$$ is sorted in non-decreasing order after all $$$n$$$ elements are inserted.

George's favorite integer is $$$7$$$. Unfortunately, $$$7$$$ is only one integer among infinitely many. Thus, George also likes integers with first digit $$$7$$$, last digit $$$7$$$, or both.

George has an array $$$a_{1...n}$$$ of $$$n$$$ integers. Count the subarrays of $$$a$$$ such that George likes the sum of the elements of the subarray. Formally, count the number of pairs $$$(i, j)$$$, where $$$1 \le i \le j \le n$$$, such that George likes $$$\displaystyle\sum_{k=i}^{j} a_k$$$.

Miyamura has baked a yummy cake for Hori-san. He wants to paint the top of the cake, which can be represented by an $$$n$$$ by $$$m$$$ grid. Unfortunately, his only paintbrush is a cube that is colored blue on two opposite sides and red on the remaining sides. Miyamura knows that Hori-san hates blueberries, so he wants to maximize the number of red cells.

Denote the upper left corner of the cake as $$$(1, 1)$$$ and the bottom right as $$$(n, m)$$$. Miyamura can choose the starting position and orientation.

In order to color the cake, Miyamura can roll the brush onto the adjacent cells such that one edge of the cubical brush touches the cake during the duration of the roll, coloring that cell of the cake with the color of the side of the brush that lands on it.

Miyamura cannot rotate the brush and stay in the same cell, and he cannot roll the paintbrush off the canvas. If he rolls on a cell that is already covered, he will replace the color of the cell with the new color.

Your task is to help him find a sequence of moves such that all cells are colored and the number of red cells is maximized.

For an array $$$x$$$, let $$$f(x)$$$ be the array obtained by deleting all $$$x_i$$$ such that there exists some $$$j \lt i$$$, and $$$x_j$$$ is equal to $$$x_i$$$, and concatenating the rest together. In other words, it keeps the first occurrence of each value and concatenates them in the order they appear in the $$$x$$$. For example, $$$f([1, 2, 1, 3]) = [1, 2, 3]$$$.

Given an array $$$a$$$, find the length of the shortest subarray $$$b$$$ of $$$a$$$ such that $$$f(a)$$$ is equal to $$$f(b)$$$.

A subarray is an array obtained by deleting several (possibly zero) elements from the beginning and end of the original array.

An array $$$c$$$ of length $$$n$$$ is good if it satisfies the following property:

• For all $$$1 \lt i \le n$$$, there exists some $$$j \lt i$$$ such that $$$\gcd(c_i, c_j) \gt 1$$$.

We can measure the cost of some array $$$b$$$ of $$$n$$$ integers. In one operation, you can remove any good subsequence from the array. The cost of $$$b$$$ is the minimum number of operations to make the array empty.

You're given an array $$$a$$$ of length $$$n$$$. You want to minimise the cost of $$$a$$$ by performing the following action exactly once:

• Choose positions $$$l$$$ and $$$r$$$ such that $$$1 \le l \le r \le n$$$.
• Perform a cyclic shift to the right on the subarray $$$a_{l \ldots r}$$$.

Please output the minimum cost after performing the following action exactly once on $$$a$$$.

Given an array $$$a$$$ with $$$n$$$ elements, $$$n = 4^p, 0 \le p \le 10, -10^9 \le a_i \le 10^9$$$, you can perform the following operation.

1. Choose an integer $$$k, 0 \le k \le p$$$.

2. Divide $$$a$$$ into $$$4^k$$$ consecutive segments of length $$$4^{p - k}$$$.

3. Choose one of these segments and either add $$$1$$$ or subtract $$$1$$$ from all the elements in that segment.

Find the minimum number of operations to make the entire array $$$0$$$.

Two best friends — Alice and Bob — are playing a Minecraft modpack when they reach the final boss. The final boss is based on the Hydra, a mythical beast which is said to grow back two heads after you chop off one of them. Luckily, the mod developers made the boss possible to beat, so it works slightly differently than that.

The hydra is initially made of $$$n$$$ heads, the $$$i$$$-th head having a strength level of $$$a_i$$$, and there is an additional parameter $$$k$$$. If Alice or Bob attacks one of the heads having $$$a_i \gt k$$$ with integer power $$$p$$$ ($$$1 \le p \le a_i$$$) — which you can choose per-attack — the strength level of the head decreases by $$$p$$$, and another head grows with $$$m$$$ strength such that $$$m$$$ is the smallest integer with $$$m^2 \ge p$$$. If any head with $$$a_i \le k$$$ is attacked, it is instantly severed and no heads regrow. Otherwise, if $$$a_i$$$ falls to zero after the attack, the head is severed, and the Hydra perishes if all heads are severed and no new ones are about to grow.

Whoever gets the final hit on the Hydra gets an achievement. Alice asks you to decide whether she should hit first or second if they both play optimally to try to get the achievement.

Given an array of positive integers, define one quicksort operation as the following:

• Choose any integer in the array as the pivot.
• Move all items strictly less than the pivot to its left, maintaining their original order.
• Move all items strictly greater than the pivot to its right, maintaining their original order.
• Any items equal to the pivot will be grouped together with it.

For example, if we have the array $$$\{5, 4, 6, 5, 3, 1, 4, 2\}$$$ and we choose the pivot to be $$$4$$$, the new array becomes $$$\{3, 1, 2, 4, 4, 5, 6, 5\}$$$.

Given an array of $$$n$$$ integers and $$$q$$$ modification queries in the form $$$a_i = v$$$, find the minimum number of quicksort operations required to sort the array into ascending order after each operation. The updates are permanent.