
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
The sequence \( a_{0}, a_{1}, \ldots \) satisfies the relationships for some parameter \( a \in \mathbf{N} \):

\[ a_{0}=0, \hspace{1mm} a_{1}=1, \hspace{1mm} a_{n+1}=2 a_{n}+(a-1) a_{n-i} \]

for \( n \in \mathrm{N} \). For a fixed prime number \( p_{0} > 2 \), find the smallest value of \( a \) such that the following two statements hold: 
a) if \( p \) is a prime number and \( p \leqslant p_{0} \), then the number \( a_{p} \) is divisible by \( p \); 
b) if \( p \) is a prime number and \( p > p_{0} \), then the number \( a_{p} \) is not divisible by \( p \).

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