
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 \( p_{0} > 2 \) be a given prime number, and the sequence \( a_{0}, a_{1}, a_{2}, \cdots \) satisfy \( a_{0} = 0 \), \( a_{1} = 1 \), and
\[ a_{n+1} = 2 a_{n} + (a - 1) a_{n-1}, \quad n = 1, 2, \cdots \]
where the natural number \( a \) is a parameter. Determine the smallest value of \( a \) that satisfies the following two conditions:
(i) If \( p \) is a prime number and \( p \leq p_{0} \), then \( p \mid a_{p} \);
(ii) If \( p \) is a prime number and \( p > p_{0} \), then \( p \times a_{p} \).

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