
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 $C:\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1(a > b > 0)$, where ${F}_{1}$, ${F}_{2}$ are its left and right focal points, and $P$ is any point on the ellipse $C$ other than the endpoints of the major axis. $G$ is a point inside $\Delta {F}_{1}P{F}_{2}$, satisfying $3\overrightarrow{PG}=\overrightarrow{P{F}_{1}}+\overrightarrow{P{F}_{2}}$, and the incenter of $\Delta {F}_{1}P{F}_{2}$ is $I$, with $\overrightarrow{IG}=\lambda \overrightarrow{{F}_{1}{F}_{2}}$ (where $\lambda$ is a real number). Find the eccentricity of the ellipse $C$, denoted as $e=$______

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