
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
Given that \( K \) is a positive integer and \( r_n \) is the remainder of \( \binom{2n}{n} \) divided by \( K \), where \( 0 \leq r_n \leq K - 1 \), find all \( K \) such that the sequence 
\[
r_1, r_2, r_3, \cdots
\]
is periodic starting from some fixed positive integer \( p \). That is, the sequence exhibits periodic behavior for all \( n \geq p \).

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