
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 \( S = \left\{ p_1 p_2 \cdots p_n \mid p_1, p_2, \ldots, p_n \text{ are distinct primes and } p_1, \ldots, p_n < 30 \right\} \). Assume 1 is in \( S \). Let \( a_1 \) be an element of \( S \). We define, for all positive integers \( n \):
\[ a_{n+1} = \frac{a_n}{n+1} \quad \text{if } a_n \text{ is divisible by } n+1; \]
\[ a_{n+1} = (n+2) a_n \quad \text{if } a_n \text{ is not divisible by } n+1. \]
How many distinct possible values of \( a_1 \) are there such that \( a_j = a_1 \) for infinitely many \( j \)'s?

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