Score : $1800$ points

Problem Statement

Given is a sequence $A$ of $N$ positive integers and a sequence $B$ of $N-1$ positive integers. You can do the following operation any number of times.

• Choose integers $i$ and $j$ ($1 \leq i < j \leq N$) and decrease each of the following values by $1$: $A_i,A_j,B_i,B_{i+1},\cdots,B_{j-1}$. Here, there should not be any negative value after this operation.

Let $m$ be the maximum number of operations that can be done. Find the number, modulo $998244353$, of sequences that $A$ can be after $m$ operations.

Constraints

• $1 \leq N \leq 2 \times 10^5$
• $1 \leq A_i \leq 10^9$
• $1 \leq B_i \leq 10^9$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$

$A_1$ $A_2$ $\cdots$ $A_N$

$B_1$ $B_2$ $\cdots$ $B_{N-1}$

Output

Print the answer.

3
1 2 2
1 2

3

We have $m=2$. After two operations, $A$ will be one of the following three sequences.

• $A=(1,0,0)$: we end up with this after operations with $(i,j)=(2,3)$ and $(i,j)=(2,3)$.
• $A=(0,1,0)$: we end up with this after operations with $(i,j)=(1,3)$ and $(i,j)=(2,3)$.
• $A=(0,0,1)$: we end up with this after operations with $(i,j)=(1,2)$ and $(i,j)=(2,3)$.

4
1 1 1 1
2 2 2

1

Note that we do not distinguish two scenarios with different contents of $B$ if the contents of $A$ are the same.

4
2 2 3 4
3 1 4

3