A15457. Sheikh (Hard Version)

This is the hard version of the problem. The only difference is that in this version $q = n$ .

You are given an array of integers $a_1, a_2, \ldots, a_n$ .

The cost of a subsegment of the array $[l, r]$ , $1 \leq l \leq r \leq n$ , is the value $f(l, r) = \operatorname{sum}(l, r) - \operatorname{xor}(l, r)$ , where $\operatorname{sum}(l, r) = a_l + a_{l+1} + \ldots + a_r$ , and $\operatorname{xor}(l, r) = a_l \oplus a_{l+1} \oplus \ldots \oplus a_r$ ( $\oplus$ stands for [bitwise XOR](https://en.wikipedia.org/wiki/Bitwise_operation#XOR)).

You will have $q$ queries. Each query is given by a pair of numbers $L_i$ , $R_i$ , where $1 \leq L_i \leq R_i \leq n$ . You need to find the subsegment $[l, r]$ , $L_i \leq l \leq r \leq R_i$ , with maximum value $f(l, r)$ . If there are several answers, then among them you need to find a subsegment with the minimum length, that is, the minimum value of $r - l + 1$ .

Each test consists of multiple test cases. The first line contains an integer $t$ ( $1 \leq t \leq 10^4$ ) — the number of test cases. The description of test cases follows.

The first line of each test case contains two integers $n$ and $q$ ( $1 \leq n \leq 10^5$ , $q = n$ ) — the length of the array and the number of queries.

The second line of each test case contains $n$ integers $a_1, a_2, \ldots, a_n$ ( $0 \leq a_i \leq 10^9$ ) — array elements.

$i$ -th of the next $q$ lines of each test case contains two integers $L_i$ and $R_i$ ( $1 \leq L_i \leq R_i \leq n$ ) — the boundaries in which we need to find the segment.

It is guaranteed that the sum of $n$ over all test cases does not exceed $2 \cdot 10^5$ .

It is guaranteed that $L_1 = 1$ and $R_1 = n$ .

For each test case print $q$ pairs of numbers $L_i \leq l \leq r \leq R_i$ such that the value $f(l, r)$ is maximum and among such the length $r - l + 1$ is minimum. If there are several correct answers, print any of them.

In all test cases, the first query is considered.

In the first test case, $f(1, 1) = 0 - 0 = 0$ .

In the second test case, $f(1, 1) = 5 - 5 = 0$ , $f(2, 2) = 10 - 10 = 0$ . Note that $f(1, 2) = (10 + 5) - (10 \oplus 5) = 0$ , but we need to find a subsegment with the minimum length among the maximum values of $f(l, r)$ . So, only segments $[1, 1]$ and $[2, 2]$ are the correct answers.

In the fourth test case, $f(2, 3) = (12 + 8) - (12 \oplus 8) = 16$ .

There are two correct answers in the fifth test case, since $f(2, 3) = f(3, 4)$ and their lengths are equal.