
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
Consider an ellipse given by the equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, where $a > b > 0$. The center of the ellipse is at the origin, with $F_1$ and $F_2$ being the left and right foci, respectively. Points $A$ and $B$ represent the upper and right vertices of the ellipse, respectively. Let $P$ be a point on the ellipse such that $PF_1$ is perpendicular to the $x$-axis and $PF_2$ is parallel to $AB$. Calculate the eccentricity of this ellipse.

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