
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
If \( m \) and \( n \) are given positive integers, a flea is on a grid in the Cartesian coordinate plane and jumps according to the following rules: if the flea is at \((x, y)\), it can only jump to one of the eight points \((x \pm m, y \pm n)\) or \((x \pm n, y \pm m)\). Suppose the initial position of the flea is \((0, 0)\). For each pair of positive integers \((m, n)\), determine whether the flea can reach \((1, 0)\) in a finite number of steps. If it can, provide the expression for the minimum number of steps \( k = f(m, n) \). If it cannot, explain why.

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