Sheikh (Hard Version)
Description
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).
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.
Input
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 $R_1 = n$.
Output
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.
6
1 1
0
1 1
2 2
5 10
1 2
2 2
3 3
0 2 4
1 3
1 2
2 3
4 4
0 12 8 3
1 4
1 3
2 4
2 3
5 5
21 32 32 32 10
1 5
1 4
1 3
2 5
3 5
7 7
0 1 0 1 0 1 0
1 7
3 6
2 5
1 4
4 7
2 6
2 7
1 1
1 1
2 2
1 1
1 1
2 2
2 3
2 3
2 3
2 3
2 3
2 3
2 3
2 3
3 4
2 4
4 6
2 4
2 4
4 6
2 4
2 4
Note
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.