
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 and assume $a \geq b$. Consider the maximum possible value of $\frac{a}{b}$, denoted as $\rho$, under the condition that there exists a solution $(x, y)$ in the equations:
\[
a^2 + y^2 = b^2 + x^2 = \sqrt{(a - x)^2 + (b - y)^2}
\]
such that $0 \leq x < a$ and $0 \leq y < b$. Additionally, assume that $(0,0)$, $(a,b-x)$, and $(b,a-y)$ form a right triangle. Find $\rho^2$.

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