
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 relatively prime positive integers \( p_{n} \) and \( q_{n} \) satisfy \(\frac{p_{n}}{q_{n}}=1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}\). Find all positive integers \( n \) such that \( 3 \mid p_{n} \).

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