OIM 2022 Problem 4

Let $n> 2$ be a positive integer. Given is a horizontal row of $n$ cells where each cell is painted blue or red. We say that a block is a sequence of consecutive boxes of the same color. Arepito the crab is initially standing at the leftmost cell. On each turn, he counts the number $m$ of cells belonging to the largest block containing the square he is on, and does one of the following:

If the square he is on is blue and there are at least $m$ squares to the right of him, Arepito moves $m$ squares to the right;

If the square he is in is red and there are at least $m$ squares to the left of him, Arepito moves $m$ cells to the left;

In any other case, he stays on the same square and does not move any further.

For each $n$ , determine the smallest integer $k$ for which there is an initial coloring of the row with $k$ blue cells, for which Arepito will reach the rightmost cell.