
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 \( a \) and \( b \) be positive real numbers with \( a \ge b \). Let \( \rho \) be the maximum possible value of \( \frac{a}{b} \) for which the system of equations
\[
a^2 + y^2 = b^2 + x^2 = (a - x)^2 + (b - y)^2
\]
has a solution in \( (x, y) \) satisfying \( 0 \le x < a \) and \( 0 \le y < b \), and also \( \angle (a-x, b-y) = 90^\circ \). Find \( \rho^2 \).

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