Leetcode 1954. Minimum Garden Perimeter to Collect Enough Apples

Given apples at each integer point (i, j) equal to |i|+|j|, find the smallest axis-aligned square centered at the origin whose contained lattice points sum to at least neededApples; this reduces to deriving a closed-form formula for the total apples inside a square of half-side k and binary-searching the minimal k (perimeter = 8·k) under large (≤1e15) inputs.