
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
Given \( x \geq y > 0 \), if there exist real numbers \( a \) and \( b \) such that \( 0 \leq a \leq x \), \( 0 \leq b \leq y \), and \( (x - a)^{2} + (y - b)^{2} = x^{2} + b^{2} = y^{2} + a^{2} \), calculate the maximum value of \( \frac{x}{y} \).

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