The subtasks are meant to reward all suboptimal solutions and are not tied to any specific methods. Here are some common constructions:

Method 1: ~\sim\left(N/2\right)^2~ pairs

Set all ~a_i = 1~.

Then, set ~a_{\left\lfloor N/2 \right\rfloor} = 10^{10}~.

Any pair ~(l,r)~ satisfying ~1 \le l \le \left\lfloor N/2 \right\rfloor \le r \le N~ is valid.

Thus, there are approximately ~\left(N/2\right)^2~ such pairs.

Method 2: ~\to N(N+1)/2~ pairs

Recurse Method 1 on both halves of the current range. The values you set for the numbers in the middle should decrease a sufficient amount (e.g. various functions work, but floor dividing by four is a simple example).

This method approaches ~N(N+1)/2~ pairs, but may run into issues for larger ~N~. Fine tuning the decreasing function may result in full points.

Intended Solution: ~N(N+1)/2~ pairs

Instead of building the array top down, we can build it bottom up. Define a recursive function f(L, R) that constructs a range of the array, ~(L, R)~, ensuring that all pairs ~(l,r)~ satisfying ~L \le l \le r \le R~ are valid. The intended solution uses the following method:

1) If ~L > R~, return.

2) If ~L = R~, set ~a_L = 1~

• ~(L,L)~ is a valid pair now. ~\checkmark~

3) Otherwise, let ~M = \lfloor(L + R) / 2\rfloor~. Recurse on f(L, M-1) and f(M+1, R).

• All pairs ~(l,r)~ satisfying ~L \le l \le r \le M-1~ or ~M+1 \le l \le r \le R~ are valid now. ~\checkmark~

4) Set ~a_M~ to ~(a_L + a_{L+1} + \dots + a_{M-1}) + (a_{M+1} + a_{M+2} + \dots + a_R) + 1~

• ~a_M~ is the minimum amount required to make all pairs ~(l,r)~ satisfying ~L \le l \le M \le r \le R~ valid. ~\checkmark~
• This covers all the remaining cases.

Thus, after calling f(L, R), all pairs satisfying ~L \le l \le r \le R~ will be valid! ~\checkmark~

We can call this recursive function on f(1, N) to receive full points. The maximum value that this construction uses when ~N = 2 \times 10^5~ is ~7380642475~, which is around ~7 \times 10^9~. This fits comfortably under the ~10^{10}~ limit.

Bonus

Python golf solution (87 bytes): https://dmoj.ca/src/6848763 (Credits to Riolku for the bulk of the logic + Josh for optimizing it with walrus operator)