
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 such that \(a^2 - 2b^2 = 0\). 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 \(0 \leq x < a\) and \(0 \leq y < b\), and such that \(x^2 + y^2 = b^2\). Find \(\rho^2\).

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