
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 \( x \geqslant y > 0 \). If there exist real numbers \( a \) and \( b \) satisfying \( 0 \leqslant a \leqslant x \), \( 0 \leqslant b \leqslant y \), and \((x-a)^2 + (y-b)^2 = x^2 + b^2 = y^2 + a^2\), determine the maximum value of \( \frac{x}{y} \).

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