Eggo has $$$n$$$ eggs, one of each size from $$$1$$$ to $$$n$$$. He wants to figure out what order to place the eggs. A powerful subarray is one where each element is one size greater than the previous. More formally, if $$$b_1, b_2, \dots, b_k$$$ are the sizes of the eggs in the subarray, then the subarray is powerful if $$$b_i - b_{i - 1} = 1$$$ for $$$2 \leq i \leq k$$$. The power of a configuration of eggs is the size of the largest powerful subarray.

You want the power of the configuration to exactly equal $$$k$$$. Can you place the eggs such that the power equal $$$k$$$? If multiple answers exist, print any.

An array $$$b$$$ is a subarray of an array $$$a$$$ if $$$b$$$ can be obtained from $$$a$$$ by deletion of several (possibly, zero or all) elements from the beginning and several (possibly, zero or all) elements from the end. Note that an array is a subarray of itself.

Oh goodness... Sean accidentally mixed up his eggs with a can of alphabet soup, and now everything's all scrambled together! Now he needs your help to rearrange the letters in the alphabet soup and restore peace to the egg-iverse.

Sean's suspicious egg-soup mixture can be represented as a series of $$$n$$$ spaghetti letters. He knows that two conditions used to be true of his letters before they got scrambled into the eggs:

1. When put in a row, his letters would form a palindrome, a string which is read the same forwards and backwards. Put more formally, if his letters were labelled $$$s_1, s_2, \dots, s_n$$$, then it is the case that $$$s_i = s_{n - i + 1}$$$ for each $$$i \in \{1, 2, \dots, n\}$$$.
2. Of all possible palindromes, his letters would form the lexicographically smallest one.

Sean knows for certain that it is possible for him to arrange the scrambled up spaghetti letters to fulfill both constraints simultaneously, but he's busy at the moment researching possible egg-spaghetti hybrids, and needs your help to unscramble his soup!

Sierra lives in the mountains where it occasionally rains egg yolk. During these storms, yolk pools in the valleys between mountain peaks. Sierra is curious about these pools and would like to know the total volume of the pool with the highest surface elevation.

The topology of the mountain range can be represented by $$$n$$$ integers, where $$$h_i$$$ describes the elevation of the $$$i$$$th location in this range. Some space above a location is considered "part of a pool" if there exist locations to the left and the right of the space with higher or equal elevations. A pool's surface height is defined by the minimum elevation of its immediate neighbors. In cases where there are ties, return the sum of the volumes of all highest pools.

Eggward is working on a hobby project and recently acquired $$$n$$$ eggs that he wants to try and hatch that are stuck at fixed positions along a number line. He has some heat lamps, and he wants to keep the eggs close to the heat lamps in order to ensure they receive the warmth they need. Unfortunately, he only has $$$k$$$ heat lamps to incubate these eggs with. His goal is to minimize the maximum distance from any egg to its nearest heat lamp. Help him find this distance. The lamps are hung above the eggs so they can be at the same position on the number line, and all lamps must be at integer positions.

What came first, the chicken or the egg? This is one of life's biggest mysteries, but your friend Chris thinks he can solve it.

He hypothesizes that whichever came first should have had more time to evolve and should thus be faster.

To test this, he has made a maze. The maze can be represented as a directed graph with $$$n$$$ nodes and $$$m$$$ edges. Among these $$$n$$$ nodes, there are $$$a$$$ entrances and $$$b$$$ exits. Because chickens and eggs are shaped differently, they take $$$c_i$$$ and $$$e_i$$$ time to traverse edge $$$i$$$, respectively.

First, Chris will test a chicken. He will uniformly randomly pick an entrance to place the chicken and time how long it takes for the chicken to reach any exit. He will then similarly pick a random entrance for the egg and time it. He plans to let the chicken and egg study the maze beforehand so that they both take their respective fastest routes to any exit.

If a chicken or egg starts at a node that can't reach an exit, it can never finish. If they both never finish, it is a tie.

Knowing you are a mathematician, Chris wants to know who you think will win.

Alice is playing a popular block game, and has $$$n$$$ days to collect as many eggs as she can. Alice initially has 1 chicken that lays an egg every day and needs 1 night of sleep. Every day, she can throw as many eggs as she wants to hatch more chickens.

Alice knows the seed of her game generated world, which gives her some additional information. In fact, she knows how many eggs she must throw to hatch another chicken, and how many nights of sleep each chicken needs before laying an egg. In fact, these values follow a predictable pattern. Help Alice find the maximum amount of eggs that she can have in $$$n$$$ days.

Note that throwing an egg requires that you have at least 1 egg, and the act of throwing an egg decreases the amount of eggs you have by 1.

Rocq the rooster owns a single egg. One day, he came back home and noticed his egg was missing! The egg is rolling around in a field — in fact, the field $$$\mathbb{Z}_p$$$, where $$$p$$$ is a prime. His egg has been studying number theory, and he knows that his egg used the following procedure to determine what path to roll away with:

1. Pick some (uniformly) random starting location $$$x_0 \in \mathbb{Z}_p$$$.
2. Pick some (uniformly) random polynomial $$$f \in \mathbb{Z}_p[x]$$$.
3. For times $$$t \gt 0$$$, the egg will roll from $$$x_{t - 1}$$$ to $$$x_t := f(x_{t - 1})$$$.

It is guaranteed the egg will eventually roll in a cycle between different locations. If Rocq can find out the expected length of the cycle, he can somehow catch the egg. Given the prime $$$p$$$, help him by finding this expectation! He doesn't like decimals, so output the answer mod $$$998244353$$$.

More formally, let $$$P = 998244353$$$. It can be shown that the answer can be represented as an irreducible fraction $$$\frac{a}{b}$$$, where $$$a$$$ and $$$b$$$ are integers and $$$b \not\equiv 0 \pmod P$$$. Output the integer $$$x$$$ such that $$$0 \leq x \lt P$$$ and $$$x \equiv a \cdot b^{-1} \pmod P$$$.

Beggsie the chicken needs you to solve a problem in Farmer John's farm. The farm can be represented as a two dimensional grid. Further, there are $$$n$$$ eggs on the farm, where the $$$i$$$th egg is at position $$$(x_i, y_i)$$$.

The compactness of the farm can be represented by the sum of Manhattan distances between each pair of eggs. Formally, if $$$(x_i', y_i')$$$ is the current position of the $$$i$$$th egg, then the compactness given by $$$\sum\limits_{i=1}^n \sum\limits_{j=i+1}^n (|x_i' - x_j'| + |y_i' - y_j'|)$$$.

There are $$$d$$$ days, and during the $$$i$$$th day there are $$$s_i$$$ seconds. During each second, Beggsie can choose to move a single egg in one of the four cardinal directions, or to not move anything. Now, let $$$c_i$$$ be the compactness of the farm after day $$$i$$$. Farmer John wants Beggsie to minimize the value of $$$\sum\limits_{i=1}^d c_i$$$. Please help Beggsie by finding the minimum such value!