
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 an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2}=1 (a > b > 0)$ with focal points $F_1$ and $F_2$ on the left and right, respectively, and a circle centered at $O$ with diameter $F_1F_2$ that intersects the ellipse at point $P$ in the first quadrant with a slope of $\sqrt{3}$ for the line $OP$, determine the eccentricity of the ellipse.

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