
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
Two players play alternately. The first player is given a pair of positive integers \((x_1, y_1)\). Each player must replace the pair \((x_n, y_n)\) that he is given by a pair of non-negative integers \((x_{n+1}, y_{n+1})\) such that \(x_{n+1} = \min(x_n, y_n)\) and \(y_{n+1} = \max(x_n, y_n) - k \cdot x_{n+1}\) for some positive integer \(k\). The first player to pass on a pair with \(y_{n+1} = 0\) wins. Find for which values of \(\frac{x_1}{y_1}\) the first player has a winning strategy.

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