# Maximize the Eligible Carpool Group Each rider i accepts a carpool only when the number of other selected riders lies in the inclusive interval [lower[i], upper[i]]. Choose any subset of riders and return the largest possible group size. ### Function Signature ```text max_eligible_riders(lower: list[int], upper: list[int]) -> int ``` ### Valid Input Domain The arrays have equal nonzero length N. Every interval is valid and bounded between zero and N minus one. ### Exact Output Semantics Return the maximum integer k for which at least k riders satisfy lower[i] <= k - 1 <= upper[i]. Return only k; the selected subset is not required, so ties among subsets do not affect the output. ### Constraints - 1 <= N <= 200,000. - 0 <= lower[i] <= upper[i] <= N - 1. - The required target time complexity is O(N). ### Public Examples #### Example 1 Input: lower = [0, 1, 1, 2, 2], upper = [1, 2, 2, 4, 4] Output: 3 At least three riders accept traveling with two other riders. #### Example 2 Input: lower = [0, 0, 0], upper = [0, 1, 2] Output: 2 Two riders accept a group with one other rider, but fewer than three accept a group with two others. ### Hints - For a proposed group size, only the number of intervals covering k minus one matters. - Use the bounded interval endpoints to avoid testing every rider against every possible k.