CF1884E.Hard Design

Consider an array of integers $b_0, b_1, \ldots, b_{n-1}$ . Your goal is to make all its elements equal. To do so, you can perform the following operation several (possibly, zero) times:

• Pick a pair of indices $0 \le l \le r \le n-1$ , then for each $l \le i \le r$ increase $b_i$ by $1$ (i. e. replace $b_i$ with $b_i + 1$ ).
• After performing this operation you receive $(r - l + 1)^2$ coins.

The value $f(b)$ is defined as a pair of integers $(cnt, cost)$ , where $cnt$ is the smallest number of operations required to make all elements of the array equal, and $cost$ is the largest total number of coins you can receive among all possible ways to make all elements equal within $cnt$ operations. In other words, first, you need to minimize the number of operations, second, you need to maximize the total number of coins you receive.

You are given an array of integers $a_0, a_1, \ldots, a_{n-1}$ . Please, find the value of $f$ for all cyclic shifts of $a$ .

Formally, for each $0 \le i \le n-1$ you need to do the following:

• Let $c_j = a_{(j + i) \pmod{n}}$ for each $0 \le j \le n-1$ .
• Find $f(c)$ . Since $cost$ can be very large, output it modulo $(10^9 + 7)$ .

Please note that under a fixed $cnt$ you need to maximize the total number of coins $cost$ , not its remainder modulo $(10^9 + 7)$ .

Each test contains multiple test cases. The first line contains the number of test cases $t$ ( $1 \le t \le 2 \cdot 10^4$ ). The description of the test cases follows.

The first line of each test case contains a single integer $n$ ( $1 \le n \le 10^6$ ).

The second line of each test case contains $n$ integers $a_0, a_1, \ldots, a_{n-1}$ ( $1 \le a_i \le 10^9$ ).

It is guaranteed that the sum of $n$ over all test cases does not exceed $10^6$ .

For each test case, for each $0 \le i \le n-1$ output the value of $f$ for the $i$ -th cyclic shift of array $a$ : first, output $cnt$ (the minimum number of operations), then output $cost$ (the maximum number of coins these operations can give) modulo $10^9 + 7$ .

In the first test case, there is only one cycle shift, which is equal to $[1]$ , and all its elements are already equal.

In the second test case, you need to find the answer for three arrays:

1. $f([1, 3, 2]) = (3, 3)$ .
2. $f([3, 2, 1]) = (2, 5)$ .
3. $f([2, 1, 3]) = (2, 5)$ .

Consider the case of $[2, 1, 3]$ . To make all elements equal, we can pick $l = 1$ and $r = 1$ on the first operation, which results in $[2, 2, 3]$ . On the second operation we can pick $l = 0$ and $r = 1$ , which results in $[3, 3, 3]$ . We have used $2$ operations, and the total number of coins received is $1^2 + 2^2 = 5$ .