A13667 | Make Them Equal

You are given an array $a$ consisting of $n$ positive integers, numbered from $1$ to $n$ . You can perform the following operation no more than $3n$ times:

1. choose three integers $i$ , $j$ and $x$ ( $1 \le i, j \le n$ ; $0 \le x \le 10^9$ );
2. assign $a_i := a_i - x \cdot i$ , $a_j := a_j + x \cdot i$ .

After each operation, all elements of the array should be non-negative.

Can you find a sequence of no more than $3n$ operations after which all elements of the array are equal?

The first line contains one integer $t$ ( $1 \le t \le 10^4$ ) — the number of test cases. Then $t$ test cases follow.

The first line of each test case contains one integer $n$ ( $1 \le n \le 10^4$ ) — the number of elements in the array. The second line contains $n$ integers $a_1, a_2, \dots, a_n$ ( $1 \le a_i \le 10^5$ ) — the elements of the array.

It is guaranteed that the sum of $n$ over all test cases does not exceed $10^4$ .

For each test case print the answer to it as follows:

- if there is no suitable sequence of operations, print $-1$ ;
- otherwise, print one integer $k$ ( $0 \le k \le 3n$ ) — the number of operations in the sequence. Then print $k$ lines, the $m$ -th of which should contain three integers $i$ , $j$ and $x$ ( $1 \le i, j \le n$ ; $0 \le x \le 10^9$ ) for the $m$ -th operation.

If there are multiple suitable sequences of operations, print any of them. Note that you don't have to minimize $k$ .