
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
An equilateral triangle $ABC$ is inscribed in an ellipse $\frac{(x-y)^2}{a^2} + \frac{(x+y)^2}{b^2} = 1$ such that vertex $A$ is at $(0, b)$, $\overline{BC}$ is parallel to the line $y = x$, and one of the foci, say $F$, is exactly at vertex $C$. Determine $\frac{AB}{b}$.

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