Lexicographically Minimal BoxIds via +1/9 Op

Given a digit string `boxIds`, repeatedly remove a digit and re-insert `min(digit + 1, 9)` anywhere in the string. Return the lex-minimal achievable string. Requirements Operation: pick any index i, remove boxIds[i], insert min(boxIds[i] + 1, 9) at any position. Apply zero or more times. Return the lexicographically smallest resulting string of the same length. Constraints: 1 <= |boxIds| <= 2 * 10^5, digits 0-9 only, leading zeros allowed. Examples boxIds = "26547" # delete 5, insert 6 -> "26647" (insert at position 4 -> "26647") # delete 6 (the original 6), insert 7 at position 4 -> "24677" output = "24677" Notes Each digit d < 9 can be "transformed" into d + 1 at the cost of moving it; digit 9 is fixed in value but can still be relocated. Greedy intuition: scan left-to-right and bubble small digits forward. Because of the increment cost, sometimes it's better to keep a 4 where it is than to promote a 5 past it; analyze the trade-off carefully. This is a non-trivial OA problem; the AI assistant in Part 2 of the same OA is the more time-consuming half, so budget tightly on Part 1. Restate the operation precisely before coding: each move is (delete at i) + (insert min(d+1, 9) anywhere). Length stays constant; digit 9 is a fixed point under the increment. Useful invariant: after any sequence of moves, the multiset of digits is determined by how many times each original digit was "promoted" (0→1→2→...→9). You are choosing both a promotion-count vector and a final permutation. Edge inputs to dry-run by hand: all 9s (no useful moves), strictly decreasing ("98765"), and repeated low digits ("00000") — these expose whether your greedy actually buys anything. Preparation Re-derive the greedy on paper for short inputs ("26547", "333", "909"). Practice the simulation in code for small inputs, then characterize when the greedy is optimal. Discuss the algorithm before coding — the interviewer/grader cares whether you can articulate the invariant, not just produce output. Before writing code, enumerate the search space for "26547" exhaustively up to 3 moves and verify the greedy matches the brute-force optimum. Articulate the invariant aloud during the screen — graders reward candidates who name the trade-off ("promoting costs a move but unlocks a smaller digit at this position") even if the final greedy is not fully proven.