2024 IMO Shortlist C4

On a board with $2024$ rows and $2023$ columns, Turbo the snail tries to move from the first row to the last row. On each attempt, he chooses to start on any cell in the first row, then moves one step at a time to an adjacent cell sharing a common side. He wins if he reaches any cell in the last row. However, there are $2022$ predetermined, hidden monsters in $2022$ of the cells, one in each row except the first and last rows, such that no two monsters share the same column. If Turbo unfortunately reaches a cell with a monster, his attempt ends and he is transported back to the first row to start a new attempt. The monsters do not move.

Suppose Turbo is allowed to take $n$ attempts. Determine the minimum value of $n$ for which he has a strategy that guarantees reaching the last row, regardless of the locations of the monsters.

在一个有 $2024$ 行、$2023$ 列的棋盘上，蜗牛 Turbo 试图从第一行走到最后一行。每次尝试时，他可以选择第一行的任意格子作为起点，然后每步移动到一个有公共边的相邻格子。若他到达最后一行的任意格子，则获胜。然而，有 $2022$ 个预先固定但隐藏的怪物分别藏在 $2022$ 个格子中，除第一行和最后一行外每行恰有一个怪物，且没有两个怪物在同一列。若 Turbo 不幸走到有怪物的格子，本次尝试结束，他被送回第一行开始新的尝试。怪物不会移动。

假设 Turbo 可以尝试 $n$ 次。求 $n$ 的最小值，使得无论怪物位置如何，他都有策略保证到达最后一行。