A12550 | 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, $$$$a[l_1]+a[l_1+1]+\dots+a[r_1]=a[l_2]+a[l_2+1]+\dots+a[r_2]= $$ $$ \dots = $$ $$ a[l_k]+a[l_k+1]+\dots+a[r_k]. $$ </li><li> 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.