
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
Find the smallest prime number $p$ that satisfies $(p, N) = 1$, where $N$ is the number of all $\left(a_{0}, a_{1}, \cdots, a_{2012}\right)$ satisfying the following conditions:
(1) $\left(a_{0}, a_{1}, \cdots, a_{2012}\right)$ is a permutation of $0, 1, \cdots, 2012$;
(2) For any positive divisor $m$ of 2013 and all $n$ such that $0 \leqslant n < n + m \leqslant 2012$, we have $m \mid \left(a_{n+m} - a_{n}\right)$.

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