
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
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. 

Present the answer in LaTex format: \boxed{Your answer}
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