
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
In the Cartesian coordinate system $xOy$, for the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$), a circle with its center at point $A$ on the ellipse is tangent to the $x$-axis at one of the ellipse's foci and intersects the $y$-axis at points $B$ and $C$. If $\triangle ABC$ is an isosceles right triangle, then the eccentricity of the ellipse is $\boxed{\frac{\sqrt{6} - \sqrt{2}}{2}}$.

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