
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
Let $F_{1}(-c,0)$ and $F_{2}(c,0)$ be the two foci of the ellipse $\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1(a > b > 0)$. Let $P$ be a point where the circle with diameter $F_{1}F_{2}$ intersects the ellipse. If $\angle PF_{1}F_{2}=2\angle PF_{2}F_{1}$, then the eccentricity of the ellipse equals \_\_\_\_\_\_.

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