C - Not Equal

Score : $300$ points

Problem Statement

You are given a sequence $C$ of $N$ integers. Find the number of sequences $A$ of $N$ integers satisfying all of the following conditions.

• $1 \leq A_i \leq C_i\, (1 \leq i \leq N)$
• $A_i \neq A_j\, (1 \leq i < j \leq N)$

Since the count may be enormous, print it modulo $(10^9+7)$.

Constraints

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

Input

Input is given from Standard Input in the following format:

$N$

$C_1$ $C_2$ $\ldots$ $C_N$

Output

Print the number of sequences $A$ of $N$ integers satisfying all of the following conditions, modulo $(10^9+7)$.

2
1 3

2

We have two sequences $A$ satisfying all of the conditions: $(1,2)$ and $(1,3)$.
On the other hand, $A=(1,1)$, for example, does not satisfy the second condition.

4
3 3 4 4

12

2
1 1

0

We have no sequences $A$ satisfying all of the conditions, so we should print $0$.

10
999999917 999999914 999999923 999999985 999999907 999999965 999999914 999999908 999999951 999999979

405924645

Be sure to print the count modulo $(10^9+7)$.