Problem Statement

There is an integer sequence $A = (A_2,A_3,\ldots,A_N)$. Also, for an integer sequence $P=(P_2, P_3, \ldots ,P_N)$ where $1 \leq P_i \leq i-1$ for each $i$ $(2 \leq i \leq N)$, define the weighted tree $T(P)$ with $N$ vertices, rooted at vertex $1$, as follows:

• A rooted tree where, for each $i$ $(2 \leq i \leq N)$, the parent of $i$ is $P_i$, and the weight of the edge between $i$ and $P_i$ is $A_i$.

You are given $Q$ queries. Process them in order. The $i$-th query is as follows:

• You are given integers $u_i$ and $v_i$, each between $1$ and $N$. For each of the possible $(N-1)!$ sequences $P$, take the tree $T(P)$ and consider the distance between vertices $u_i$ and $v_i$ in this tree. Output the sum, modulo $998244353$, of these distances over all $T(P)$. Here, the distance between two vertices $u_i$ and $v_i$ is the sum of the weights of the edges on the unique path (not visiting the same vertex more than once) that connects them.

Constraints

• $2 \leq N \leq 2 \times 10^5$
• $1 \leq Q \leq 2 \times 10^5$
• $1 \leq A_i \leq 10^9$
• $1 \leq u_i < v_i \leq N$
• All input values are integers.

Input

The input is given from Standard Input in the following format:

N Q
A_2 A_3 \ldots A_N
u_1 v_1
u_2 v_2
\vdots
u_Q v_Q

Output

Print $Q$ lines. The $i$-th line should contain the answer to the $i$-th query.