CF1141F2 Same Sum Blocks (Hard)

This problem is given in two editions, which differ exclusively in the constraints on the number $ n $ . You are given an array of integers $ a[1], a[2], \dots, a[n]. $ A block is a sequence of contiguous (consecutive) elements $ a[l], a[l+1], \dots, a[r] $ ( $ 1 \le l \le r \le n $ ). Thus, a block is defined by a pair of indices $ (l, r) $ . Find a set of blocks $ (l_1, r_1), (l_2, r_2), \dots, (l_k, r_k) $ such that: - They do not intersect (i.e. they are disjoint). Formally, for each pair of blocks $ (l_i, r_i) $ and $ (l_j, r_j $ ) where $ i \neq j $ either $ r_i < l_j $ or $ r_j < l_i $ . - For each block the sum of its elements is the same. Formally, $$ a[l_{1}] + a[l_{1} + 1] + \dots + a[r_{1}] = a[l_{2}] + a[l_{2} + 1] + \dots + a[r_{2}] = \dots = a[l_{k}] + a[l_{k} + 1] + \dots + a[r_{k}] $$ - The number of the blocks in the set is maximum. Formally, there does not exist a set of blocks $ (l\_1', r\_1'), (l\_2', r\_2'), \dots, (l\_{k'}', r\_{k'}') $ satisfying the above two requirements with $ k' > k$. The picture corresponds to the first example. Blue boxes illustrate blocks.Write a program to find such a set of blocks.

The first line contains integer $ n $ ( $ 1 \le n \le 1500 $ ) — the length of the given array. The second line contains the sequence of elements $ a[1], a[2], \dots, a[n] $ ( $ -10^5 \le a_i \le 10^5 $ ).

In the first line print the integer $ k $ ( $ 1 \le k \le n $ ). The following $ k $ lines should contain blocks, one per line. In each line print a pair of indices $ l_i, r_i $ ( $ 1 \le l_i \le r_i \le n $ ) — the bounds of the $ i $ -th block. You can print blocks in any order. If there are multiple answers, print any of them.