
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
Given the ellipse $\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1\ (a > b > 0)$, with $F\_{1}$ as the left focus, $A$ as the right vertex, and $B\_{1}$, $B\_{2}$ as the upper and lower vertices respectively. If the four points $F\_{1}$, $A$, $B\_{1}$, and $B\_{2}$ lie on the same circle, find the eccentricity of this ellipse.

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