In a grid of $100\times 100$ squares, there is a mouse on the top-left square, and there is a piece of cheese in the bottom-right square. The mouse wants to move to the bottom-right square to eat the cheese. For each step, the mouse can move from one square to an adjacent square (two squares are considered adjacent if they share a common edge). Now, any divider can be placed on the common edge of two adjacent squares such that the mouse cannot directly move between these two adjacent squares. A placement of dividers is called "kind" if the mouse can still reach the cheese after the dividers are placed. Find the smallest positive integer $n$ such that, regardless of any "kind" placement of $2023$ dividers, the mouse can reach the cheese in at most $n$ steps.