
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 $\sigma$ be the maximum possible value of $\frac{b}{a}$ for which the system of equations
$$
b^2 + x^2 = a^2 + y^2 = (b - x)^2 + (a - y)^2
$$
has a solution in $(x,y)$ satisfying $0 \le x < b$ and $0 \le y < a$. Find $\sigma^2$.

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