
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 number 12320 is written on the board. Petya appended $10n + 1$ threes to it, where $n$ is a nonnegative integer. Vasya thought this was a base-4 number representing a natural number $x$, and he factored $x$ into prime factors. It turned out that there were exactly two distinct prime factors among them. For which $n$ is this possible?

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