
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 $0 \le y < b$, where $(a, y)$, $(x, b)$, and $(0, 0)$ form a right triangle with right angle at $(0,0)$. Find $\rho^2$.

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