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:
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:
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 $$$998244353$$$.
6 4 <?>?> 1 ? 4 < 5 < 1 >
3 0 0 0 1
2 2 > 1 ? 1 <
1 0 1
In the first example, there are three possible orderings before all queries:
After the last query, there is only one possible ordering:
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