
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 positive integers. A circular necklace contains \( m n \) beads, each either red or blue. It turned out that no matter how the necklace was cut into \( m \) blocks of \( n \) consecutive beads, each block had a distinct number of red beads. Determine, with proof, all possible values of the ordered pair \((m, n)\).

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