
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 \( m \) and \( n \) be relatively prime positive integers. The numbers \( \frac{m}{n} \) and \( \frac{n}{m} \) are written on a blackboard. At any point, Evan may pick two of the numbers \( x \) and \( y \) written on the board and write either their arithmetic mean \( \frac{1}{2}(x+y) \) or their harmonic mean \( \frac{2xy}{x+y} \). For which \( (m, n) \) can Evan write 1 on the board in finitely many steps?

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