
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 $E$, with $A$ being the left vertex, and $P$ is a point on the ellipse $E$ such that the circle with diameter $PF_1$ passes through $F_2$ and $|PF_{2}|= \frac {1}{4}|AF_{2}|$, determine the eccentricity of the ellipse $E$.

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