
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 $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, let $P$ be a point on the ellipse such that the projection of $P$ onto the $x$-axis is the left focus $F_1$. Let $A$ be the intersection point of the ellipse with the positive semi-major axis, and let $B$ be the intersection point of the ellipse with the positive semi-minor axis, such that $AB$ is parallel to $OP$ (where $O$ is the origin). Find the eccentricity of the ellipse.

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