CF1732C2 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.