Score : $1000$ points

Problem Statement

Given is an integer sequence of length $N+1$: $X_0,X_1,\ldots,X_N$, where $0=X_0 < X_1 < \ldots < X_N$ holds.

Now, $N$ people numbered $1$ through $N$ will appear on a number line. Person $i$ will appear at a real coordinate chosen uniformly at random from the interval $[X_{i-1},X_i]$.

Find the expected value of the smallest distance between two people, modulo $998244353$.

Constraints

• $2 \leq N \leq 20$
• $0=X_0 < X_1 < \cdots < X_N \leq 10^6$

Input

Input is given from Standard Input in the following format:

$N$

$X_0$ $X_1$ $\ldots$ $X_N$

Output

Print the expected value of the smallest distance between two people, modulo $998244353$.

2
0 1 3

499122178

There are just two people, so the expected value of the smallest distance between two people is just the expected value of the distance between Person $1$ and Person $2$. The answer is $3/2$.

5
0 3 4 8 9 14

324469854

The answer is $196249/172800$.

20
0 38927 83112 125409 165053 204085 246405 285073 325658 364254 406395 446145 485206 525532 563762 605769 644863 683453 722061 760345 798556

29493181