Not Quite Lee
Description
Lee couldn't sleep lately, because he had nightmares. In one of his nightmares (which was about an unbalanced global round), he decided to fight back and propose a problem below (which you should solve) to balance the round, hopefully setting him free from the nightmares.
A non-empty array $b_1, b_2, \ldots, b_m$ is called good, if there exist $m$ integer sequences which satisfy the following properties:
- The $i$-th sequence consists of $b_i$ consecutive integers (for example if $b_i = 3$ then the $i$-th sequence can be $(-1, 0, 1)$ or $(-5, -4, -3)$ but not $(0, -1, 1)$ or $(1, 2, 3, 4)$).
- Assuming the sum of integers in the $i$-th sequence is $sum_i$, we want $sum_1 + sum_2 + \ldots + sum_m$ to be equal to $0$.
You are given an array $a_1, a_2, \ldots, a_n$. It has $2^n - 1$ nonempty subsequences. Find how many of them are good.
As this number can be very large, output it modulo $10^9 + 7$.
An array $c$ is a subsequence of an array $d$ if $c$ can be obtained from $d$ by deletion of several (possibly, zero or all) elements.
The first line contains a single integer $n$ ($2 \le n \le 2 \cdot 10^5$) — the size of array $a$.
The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le 10^9$) — elements of the array.
Print a single integer — the number of nonempty good subsequences of $a$, modulo $10^9 + 7$.
Input
The first line contains a single integer $n$ ($2 \le n \le 2 \cdot 10^5$) — the size of array $a$.
The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le 10^9$) — elements of the array.
Output
Print a single integer — the number of nonempty good subsequences of $a$, modulo $10^9 + 7$.
Samples
4
2 2 4 7
10
10
12391240 103904 1000000000 4142834 12039 142035823 1032840 49932183 230194823 984293123
996
Note
For the first test, two examples of good subsequences are $[2, 7]$ and $[2, 2, 4, 7]$:
For $b = [2, 7]$ we can use $(-3, -4)$ as the first sequence and $(-2, -1, \ldots, 4)$ as the second. Note that subsequence $[2, 7]$ appears twice in $[2, 2, 4, 7]$, so we have to count it twice.
Green circles denote $(-3, -4)$ and orange squares denote $(-2, -1, \ldots, 4)$. For $b = [2, 2, 4, 7]$ the following sequences would satisfy the properties: $(-1, 0)$, $(-3, -2)$, $(0, 1, 2, 3)$ and $(-3, -2, \ldots, 3)$