
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
The ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ (where $a > b > 0$) has foci $F_{1}$ and $F_{2}$ located on the left and right ends respectively, and a point $P$ on the ellipse which does not coincide with the major axis vertices. Points $I$ and $G$ are the incenter and centroid of $\triangle P F_{1} F_{2}$ respectively. Given that for any point $\boldsymbol{P}$, $\boldsymbol{I}$, and $\boldsymbol{G}$ are always perpendicular to the $\boldsymbol{x}$-axis, find the eccentricity of the ellipse.

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