
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 $c$ and $d$ be positive real numbers with $c \geq d$. Let $\sigma$ be the maximum possible value of $\frac{c}{d}$ for which the system of equations $$
c^2 + y^2 = d^2 + x^2 = (c - x)^2 + (d - y)^2
$$ has a solution in $(x,y)$ satisfying $0 \leq x < c$ and $0 \leq y < d$. Find $\sigma^2$.

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