
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 coprime positive integers. Initially, the numbers \(\frac{m}{n}\) and \(\frac{n}{m}\) are written on the blackboard. Next, the following operation is repeatedly performed: Select any two existing numbers \( x \) and \( y \), and then write either \(\frac{x+y}{2}\) or \(\frac{2xy}{x+y}\) on the blackboard. Determine all values of \( m \) and \( n \) such that it is possible to make the number 1 appear on the blackboard after a finite number of operations.

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