
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 $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ $(a > b > 0)$ with its left and right foci respectively as $F_1$ and $F_2$, and a point $P$ on the ellipse, with $O$ being the origin, find the eccentricity of the ellipse given that $|OP| = \dfrac{1}{2}|F_1F_2|$ and $|PF_1||PF_2| = a^2$.

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