
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 the focus of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)$ be $F_{1}$, $F_{2}$, and $P$ be a point on the ellipse with $\angle F_1PF_2=\frac{π}{3}$. If the radii of the circumcircle and incircle of $\triangle F_{1}PF_{2}$ are $R$ and $r$ respectively, and when $R=3r$, the eccentricity of the ellipse is ______.

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