
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 a sequence \( \{ a_n \}_{n=0}^{\infty} \) be defined by \( a_0 = \sqrt{2} \), \( a_1 = 2 \), and \( a_{n+1} = a_n a_{n-1}^2 \) for \( n \geq 1 \). The sequence of remainders when \( a_0, a_1, a_2, \ldots \) are divided by 2014 is eventually periodic with some minimal period \( p \) (meaning that \( a_m = a_{m+p} \) for all sufficiently large integers \( m \), and \( p \) is the smallest such positive integer). Find \( p \).

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