There are $N$ plates numbered $1$ through $N$. Dish $i$ has $a_i$ stones on it. There is also an empty bag.  
You can perform the following two kinds of operations any number of times (possibly zero) in any order.

-   Remove one stone from each plate with one or more stones. Put the removed stones into the bag.
-   Take $N$ stones out of the bag, and put one stone to each plate. This operation can be performed only when the bag has $N$ or more stones.

Let $b_i$ be the number of stones on plate $i$ after you finished the operations. Print the number, modulo $998244353$, of sequences of integers $(b_1, b_2, \dots, b_N)$ of length $N$ that can result from the operations.

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

```
$N$
$a_1$ $a_2$ $\dots$ $a_N$
```

Print the number, modulo $998244353$, of possible sequences $(b_1, b_2, \dots, b_N)$.

-   $1 \leq N \leq 2 \times 10^5$
-   $0 \leq a_i \leq 10^9$

7

For example, b becomes (2,1,2) by the following procedure.

• Perform the first operation. b becomes (2,0,2).
• Perform the first operation. b becomes (1,0,1).
• Perform the second operation. b becomes (2,1,2).

The following seven sequences can be the resulting b.

• (0,0,0)
• (1,0,1)
• (1,1,1)
• (2,0,2)
• (2,1,2)
• (2,2,2)
• (3,1,3)

1

There are one sequence, (0), that can be the resulting b.