
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 \(a \ge b\). Consider the system of equations:
\[ a^2 + y^2 = b^2 + x^2 = (a - x)^2 + (b + y)^2 \]
with the condition that there exists a solution \((x, y)\) such that \(0 \leq x < a\) and \(0 \leq y < b\). Determine the maximum possible value of \(\frac{a}{b}\), denoted as \(\rho\), and find \(\rho^2\).

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