
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 $a\ge b$. Determine the maximum possible value of $\frac {a}{b}$ for which the system of equations $$
a^2 + z^2 = b^2 + x^2 = (a - x)^2 + (b - z)^2
$$ has a solution in $(x, z)$ satisfying $0\le x < a$ and $0\le z < b$. Find $\rho^2$, where $\rho$ is this maximum value.

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