A16360 | Monocarp and the Set

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 $998244353$ .

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 ?).

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 $998244353$ .