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문제

For an array $[b_1, b_2, \dots , b_m]$ of integers, let’s define its weight as the sum of pairwise products of its elements, namely as the sum of $b_ib_j$ over $1 \le i < j \le m$.

You are given an array of $n$ integers $[a_1, a_2, \dots , a_n]$, and are asked to find the number of pairs of integers $(l, r)$ with $1 \le l \le r \le n$, for which the weight of the subarray $[a_l, a_{l+1}, \dots , a_r]$ is divisible by $3$.

입력

The first line of the input contains a single integer $n$ ($1 \le n \le 5 \cdot 10^5$) — the length of the array.

The second line contains $n$ integers $a_1, a_2, \dots , a_n$ ($0 \le a_i \le 10^9$) — the elements of the array.

출력

Output a single integer — the number of pairs of integers $(l, r)$ with $1 \le l \le r \le n$, for which the weight of the corresponding subarray is divisible by $3$.

예제 입력 1

3
5 23 2021

예제 출력 1

4

예제 입력 2

5
0 0 1 3 3

예제 출력 2

15

예제 입력 3

10
0 1 2 3 4 5 6 7 8 9

예제 출력 3

20

힌트

In the first sample, the weights of exactly $4$ subarrays are divisible by $3$:

  • weight($[5]$) = weight($[23]$) = weight($[2021]$) = $0$
  • weight($[5, 23, 2021]$) = $56703$ = $3 \cdot 41 \cdot 461$