Minimum Steps to Reach a Target with Variable Jumps

For the minimum number of jumps when a[i] is the maximum forward distance, scan once while tracking farthest, the greatest index reachable using the current number of jumps or fewer, and end, the boundary of the current jump layer. For each index before the last, update farthest = max(farthest, i + a[i]). When i reaches end, every position reachable with the current jump count has been considered, so increment jumps and set end to farthest. This greedy choice is safe because choosing an earlier landing cannot expose an index beyond the maximum frontier already observed. If farthest does not advance at a boundary before reaching the end, the target is unreachable and return the specified failure value. Handle length zero or one as zero jumps, a zero at the start, and values extending beyond n-1. The approach is O(n) time and O(1) space; do not confuse it with the easier yes/no Jump Game.