
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
For a given prime \( p \), find the number of different sequences of positive integers \( a_{0}, a_{1}, a_{2}, \cdots \) that satisfy
$$
\frac{a_{0}}{a_{1}}+\frac{a_{0}}{a_{2}}+\cdots+\frac{a_{0}}{a_{n}}+\frac{p}{a_{n+1}}=1, \quad n=1, 2, \cdots
$$

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