
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 that $F_{1}$ and $F_{2}$ are the left and right foci of the ellipse $E$, respectively, $A$ is the left vertex, and $P$ is a point on the ellipse $E$, a circle with diameter $PF_{1}$ passes through $F_{2}$. If $|PF_{2}|= \frac {1}{4}|AF_{2}|$, then calculate the eccentricity of the ellipse $E$.

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