2022 AIME II Problem 6

Let x_1\leq x_2\leq \cdots\leq x_{100} be real numbers such that |x_1| + |x_2| + \cdots + |x_{100}| = 1 and x_1 + x_2 + \cdots + x_{100} = 0. Among all such 100-tuples of numbers, the greatest value that x_{76} - x_{16} can achieve is \tfrac mn, where m and n are relatively prime positive integers. Find m+n.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.