Valid Triangle Number

For positive sorted sides a <= b <= c, only a+b>c needs checking. Sort the array, then fix c at each index from right to left. Set left at zero and right at c-1. If nums[left]+nums[right] > nums[c], every index from left through right-1 paired with right also works because sorted values are at least nums[left], so add right-left and decrement right. Otherwise increment left to seek a larger sum. Sorting costs O(n log n) and the nested pointer sweeps cost O(n^2), with O(1) extra space beyond sorting. Use a wide sum if values can overflow. Skip or naturally reject zero lengths. Test duplicates, all equal values, arrays shorter than three, and a near-degenerate triple where sum equals c.