
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  $2561$  given points on a circle be colored either red or green. In each step, all points are recolored simultaneously in the following way: if both direct neighbors of a point  $P$  have the same color as  $P$ , then the color of  $P$  remains unchanged, otherwise  $P$  obtains the other color. Starting with the initial coloring  $F_1$ , we obtain the colorings  $F_2, F_3,\dots$  after several recoloring steps. Determine the smallest number  $n$  such that, for any initial coloring  $F_1$ , we must have  $F_n = F_{n+2}$ .

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