
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 $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a>b>0)$ with left and right foci $F_1$ and $F_2$, respectively. A line passing through $F_1$ intersects the ellipse at points $A$ and $B$, with $AF_2$ perpendicular to the x-axis. Let $P$ be a moving point on the ellipse, distinct from $A$ and $B$, such that $S_{△PAB}=4S_{△PBF_{1}}$. Determine the eccentricity of the ellipse.

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