
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 an ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$ with left and right foci $F_{1}(-c,0)$ and $F_{2}(c,0)$ respectively, and a point $P$ on the ellipse (different from the left and right vertices), the radius of the inscribed circle of $\triangle PF_{1}F_{2}$ is $r$. If the maximum value of $r$ is $\frac{c}{3}$, then the eccentricity of the ellipse is ______.

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