
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 $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with left and right foci $F\_1$, $F\_2$. Let $P$ be a point on the ellipse, and $\triangle F\_1 P F\_2$ have centroid $G$ and incenter $I$. If $\overrightarrow{IG} = λ(1,0) (λ ≠ 0)$, find the eccentricity $e$ of the ellipse.

A) $\frac{1}{2}$
B) $\frac{\sqrt{2}}{2}$
C) $\frac{1}{4}$
D) $\frac{\sqrt{5}-1}{2}$

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