
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 the ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, $F_1$ and $F_2$ are its left and right foci, respectively, and $P$ is any point on the ellipse $C$ other than the endpoints of the major axis. Let $G$ be a point inside $\triangle F_1PF_2$ such that $3 \overrightarrow{PG} = \overrightarrow{PF_1} + \overrightarrow{PF_2}$. The incenter of $\triangle F_1PF_2$ is $I$, and it is given that $\overrightarrow{IG} = \lambda \overrightarrow{F_1F_2}$ (where $\lambda$ is a real number). Find the eccentricity $e$ of the ellipse $C$.

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