1886D · Monocarp and the Set

Problem: Monocarp has $n$ numbers $1, 2, \dots, n$ and a set (initially empty). He adds his numbers to this set $n$ times in some order. During each step, he adds a new number (which has not been present in the set before). In other words, the sequence of added numbers is a permutation of length $n$.

Every time Monocarp adds an element into the set except for the first time, he writes out a character:

• if the element Monocarp is trying to insert becomes the maximum element in the set, Monocarp writes out the character >;
• if the element Monocarp is trying to insert becomes the minimum element in the set, Monocarp writes out the character <;
• if none of the above, Monocarp writes out the character ?.

You are given a string $s$ of $n-1$ characters, which represents the characters written out by Monocarp (in the order he wrote them out). You have to process $m$ queries to the string. Each query has the following format:

• $i$ $c$ — replace $s_i$ with the character $c$.

Both before processing the queries and after each query, you have to calculate the number of different ways to order the integers $1, 2, 3, \dots, n$ such that, if Monocarp inserts the integers into the set in that order, he gets the string $s$. Since the answers might be large, print them modulo $998244353$.

Input Format: The first line contains two integers $n$ and $m$ ($2 \le n \le 3 \cdot 10^5$; $1 \le m \le 3 \cdot 10^5$).

The second line contains the string $s$, consisting of exactly $n-1$ characters <, > and/or ?.

Then $m$ lines follow. Each of them represents a query. Each line contains an integer $i$ and a character $c$ ($1 \le i \le n-1$; $c$ is either <, >, or ?).

Output Format: Both before processing the queries and after each query, print one integer — the number of ways to order the integers $1, 2, 3, \dots, n$ such that, if Monocarp inserts the integers into the set in that order, he gets the string $s$. Since the answers might be large, print them modulo $998244353$.

Note: In the first example, there are three possible orderings before all queries:

• $3, 1, 2, 5, 4, 6$;
• $4, 1, 2, 5, 3, 6$;
• $4, 1, 3, 5, 2, 6$.

After the last query, there is only one possible ordering:

• $3, 5, 4, 6, 2, 1$.