Aryan discovered that he might be at risk of Jeffersonoma, a devastating cellular disease, because his cells may carry out mitosis incorrectly. Mitosis is the process by which a cell splits into two daughter cells.

Specifically, he is concerned about the possibility that the chromosomes in his cells may not split properly. He has decided to measure the number of chromosomes of a cell sample before and after mitosis.

While normal human cells have $$$46$$$ chromosomes, Aryan's cells have $$$c$$$ chromosomes, where $$$c$$$ is positive integer. Aryan observes his sample cell splits into two daughter cells with $$$a$$$ and $$$b$$$ chromosomes respectively. The cell has correctly split if and only if $$$a$$$ and $$$b$$$ add up to $$$c$$$. If not, Aryan may be at risk of Jeffersonoma.

Aryan gives you the values of $$$a$$$, $$$b$$$, and $$$c$$$. Determine if he is not at risk for Jeffersonoma.

A Neural Network is a powerful Machine Learning technique that is inspired by the human brain. A Neural Network consists of several layers of nodes where information is passed through edges connecting the nodes.

More specifically, Akash is working with a Neural Network with $$$n$$$ layers, where the $$$i$$$-th layer has $$$a_i$$$ nodes. For every two adjacent layers, there is an edge connecting all pairs of nodes between one layer to the other layer. In other words, if two layers have $$$x$$$ and $$$y$$$ nodes, respectively, then a total of $$$x \cdot y$$$ edges will connect those layers.

Given the structure of his Neural Network, find the total number of edges needed to connect each pair of adjacent layers.

An organism's DNA can be represented by a string consisting of the characters "A", "T", "C", and "G". Due to the structure of DNA, "A" pairs with "T" and "C" pairs with "G", and these are called complementary pairs.

Daniel has sequenced a specific gene from an organism and has determined its DNA corresponds to the string $$$X$$$ with length $$$n$$$. He wants to compare this string to the target DNA $$$Y$$$ which is also a string of length $$$n$$$.

Daniel knows that because of mutations, the strings $$$X$$$ and $$$Y$$$ may not be equal. However, he hypothesizes that the only type of mutation that $$$X$$$ has is swapping complementary pairs. For example, a character "G" in the $$$i$$$-th location in $$$Y$$$ may have mutated into a character "C" at that same location but not a character "A", since "A" and "G" are not complementary.

Given both strings $$$X$$$ and $$$Y$$$, help Daniel determine whether his hypothesis is correct.

This is the easy version of the problem. The only difference between the easy and hard versions is the constraint on $$$n$$$. Solutions that pass under the time limit for the easy version may not pass the hard version.

Sritan has recently been investigating the predatory relationships of the TJ ecosystem. He has discovered that the $$$i$$$-th animal can be given a predator factor of $$$p_i$$$, which is a positive integer.

He learns that animals will prey on other animals based on their relative predator factors. An animal $$$i$$$ will prey on animal $$$j$$$ if the difference between $$$p_i$$$ and $$$p_j$$$ is at least $$$a$$$. However, if this difference is $$$b$$$ or more, animal $$$i$$$ considers the animal $$$j$$$ to be too weak and will not eat it.

There are $$$n$$$ animals in the TJ ecosystem. For each animal, find the number of predators and prey that it has.

This is the hard version of the problem. The only difference between the easy and hard versions is the constraint on $$$n$$$. Solutions that pass under the time limit for the easy version may not pass the hard version.

Sritan has recently been investigating the predatory relationships of the TJ ecosystem. He has discovered that the $$$i$$$-th animal can be given a predator factor of $$$p_i$$$, which is a positive integer.

He learns that animals will prey on other animals based on their relative predator factors. An animal $$$i$$$ will prey on animal $$$j$$$ if the difference between $$$p_i$$$ and $$$p_j$$$ is at least $$$a$$$. However, if this difference is $$$b$$$ or more, animal $$$i$$$ considers the animal $$$j$$$ to be too weak and will not eat it.

There are $$$n$$$ animals in the TJ ecosystem. For each animal, find the number of predators and prey that it has.

Jasmine has recently learned that the mitochondria are the powerhouse of the cell and produce ATP, which is the energy currency of your body. Since Jasmine is healthy and fit, her mitochondria produce $$$m$$$ ATP each day.

Jasmine has to do $$$n$$$ tasks today, numbered from $$$1$$$ to $$$n$$$ from highest to lowest priority. The $$$i$$$-th task costs $$$a_i$$$ ATP. Since some tasks have higher priority than others, Jasmine will choose some integer $$$k$$$ and only consider doing the $$$k$$$ tasks with the highest priority. Out of the tasks she considers, she tries to complete the most possible tasks. Jasmine cannot do the same task more than once.

For each $$$k$$$ from $$$1$$$ to $$$n$$$, find the maximum number of tasks that Jasmine could complete if she only considers the $$$k$$$ tasks with the highest priority.

Sanchali is a researcher who wants to protect an endangered species of trees. Their habitat consists of $$$n$$$ forests arranged in a line, where the $$$i$$$-th forest has a density of $$$t_i$$$.

Unfortunately, this habitat is very vulnerable to wildfires. A forest is either healthy or is burning with a wildfire of some integer strength $$$k \gt 0$$$. Wildfires can spread to adjacent forests (forests $$$i$$$ and $$$j$$$ are adjacent if $$$\lvert i - j \rvert = 1$$$).

For a pair of adjacent forests $$$i$$$ and $$$j$$$, if forest $$$i$$$ is healthy but forest $$$j$$$ has a wildfire of strength $$$k$$$, then a wildfire of strength $$$k'$$$ starts in forest $$$i$$$, where

$$$k' = \begin{cases} k+1 & k \lt t_i \\ k & k = t_i \\ k-1 & k \gt t_i \\ \end{cases} $$$

If $$$k' = 0$$$, no fire starts.

For each $$$i$$$ from $$$1$$$ to $$$n$$$, find the minimum strength needed for a wildfire to start at forest $$$i$$$ and burn all of the $$$n$$$ forests in the habitat.

Dr. Zarif is studying the genome of a certain prokaryote and has developed an ingenious gene editing technology called the Lateral Omni-genomic Precision Integration and Targeting Apparatus and Laboratory (LOPITAL) to assist him.

An RNA strand is a string of nucleotides that are represented by one of the characters "A", "U", "C", or "G" which can be either circular or linear. For the organism that he is currently studying, Dr. Zarif is more interested in the number of linear RNA strands and doesn't care about circular RNA.

Dr. Zarif is planning to carry out some tests on the organism's genome, which is initially a single circular strand of RNA with $$$n$$$ nucleotides. Using LOPITAL, Dr. Zarif will do $$$q$$$ cumulative operations on the RNA, where each operation is one of the following types:

• CUT: LOPITAL cuts all RNA strands between every consecutive occurrence of nucleotides $$$x$$$ and $$$y$$$. As a result of this operation, circular RNA strands may become linear, and linear RNA may separate into two or more different strands.

• SPLICE: LOPITAL splices two different RNA strands wherever one of them ends with nucleotide $$$x$$$ and the other begins with nucleotide $$$y$$$. Only linear RNA can be spliced since circular RNA does not have a beginning or an end. LOPITAL can splice strands in any order but will keep splicing until no strands satisfy the above condition.

Note that since RNA strands are not bidirectional, cuts between $$$x$$$ and $$$y$$$ are not the same as between $$$y$$$ and $$$x$$$, and similarly for splices.

Dr. Zarif is interested in knowing the maximum possible number of linear RNA strands that could exist after each operation. Given the nucleotide sequence of the original circular RNA strand and the $$$q$$$ operations he performs, help Dr. Zarif compute these values.

Dr. Vijayram has an infinite grid of agar to grow bacteria. This grid can be represented by the $$$xy$$$-plane where bacteria can grow at integer coordinates. Every minute, each living bacteria on the grid can spread to any number of adjacent locations$$$^\dagger$$$ (including not spreading at all). A bacteria might also die and be decomposed after spreading.

To prepare for future experiments, Dr. Vijayram had sterilized the grid. Unfortunately, the grid was contaminated $$$t$$$ minutes ago by a bacteria at the point $$$(x_0, y_0)$$$. Dr. Vijayram doesn't know what $$$t$$$, $$$x_0$$$, and $$$y_0$$$ are, but he can tell you the locations of all bacteria that are currently alive. Help him find the lexicographically minimum valid tuple $$$(t, x_0, y_0)$$$ that is consistent with Dr. Vijayram's observations.

$$$^\dagger$$$Two points $$$(x_1, y_1)$$$ and $$$(x_2, y_2)$$$ are adjacent if $$$\lvert x_1 - x_2 \rvert + \lvert y_1 - y_2 \rvert = 1$$$.

Medha is a researcher who is investigating a certain species of monkeys. These species have two characteristics of interest to researchers: a fitness score (represented by a positive integer) and a special OP gene. The OP gene has the property that at any given time, it is present in exactly one species of monkeys.

At the beginning of every epoch, a new species of monkey evolves and branches off from each existing species, but the original species also continues to exist. The only exception is the species that currently contains the OP gene, in which case a new species may or may not evolve from it.

Let's call the new species the child and the species it evolved from the parent. Since evolution is very effective at producing beneficial adaptations, the child species will always have a higher fitness score than its parent species.

In the present day, there are $$$n$$$ different species of monkeys where the $$$i$$$-th species has a fitness score of $$$i$$$. It is known that all species evolved at exactly one time in history and evolved from one parent species. Species $$$1$$$ is an ancestor of all other species.

Count the number of distinct evolutionary pathways modulo $$$998244353$$$. Two pathways are distinct if there are two species with the same fitness score that evolved from different parent species.