
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 1200 is written on the board. Petl appended $10 n+2$ fives to it, where $n$ is a non-negative integer. Vasya interpreted this as the representation of a natural number $x$ in base 6 and factored $x$ into prime factors. It turned out that there were exactly two distinct prime factors. For which values of $n$ is this possible?

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