
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
We have an a sequence such that  $a_n = 2 \cdot 10^{n + 1} + 19$ . Determine all the primes  $p$ , with  $p \le 19$ , for which there exists some  $n \ge 1$  such that  $p$  divides  $a_n$ .

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