Recently brz has fallen for polynomials. He thinks if a polynomial $$$f(x)$$$ satisfies $$$f(a)=b$$$, it would be cool!

One day Mandy buy him a magic sequence $$$c$$$. He can't wait to apply his new polynomial theory to the sequence. Let polynomial $$$f(x)=a_0+a_1x+a_2x^2+\cdots$$$. He defines $$$G(f(x))$$$ as the number of pairs $$$(i,j)$$$ satisfying the following conditions:

1. $$$1\le i,j\le n$$$,$$$i\neq j$$$
2. $$$f(c_i)=c_j$$$
3. for all non-negative integer $$$k$$$, $$$a_k \lt c_i$$$.

Now he wants to know the sum of $$$G(f(x))$$$ where $$$f(x)$$$ satisfies that for all non-negative integer $$$k$$$, $$$a_k\geq 0,a_k\in \mathbb{Z}$$$.

However, the magic sequence can change itself. It will change one single element in it by $$$m$$$ times. brz wants to know the sum of $$$G(f(x))$$$ after every operation of the sequence.

Formally speaking, Let's say set $$$S$$$ contains all the polynomial that $$$\forall k,a_k\geq 0,a_k\in \mathbb{Z}$$$. You need to calculate $$$\sum_{f(x)\in S} G(f(x))$$$. Note that $$$|S|=\infty$$$.

Since the answer may be too large, you only need to find the answer module $$$998~244~353$$$.