Implement lower_bound on unknown-size sorted array

Given a non-decreasing integer array A of unknown length, accessible only via API get(i) that returns +∞ for i ≥ n, implement lower_bound(target): the smallest index i with A[i] ≥ target, or −1 if none exists. Use exponential search to bracket n and then binary search to find the answer with overall O(log n) time. Handle duplicates, empty array, all values < target, and all values ≥ target. Prove correctness, give a tight upper bound on the number of get calls, and explain how your approach changes if A may be rotated once (with or without duplicates).