
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
Let point $P$ be a point on the ellipse $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$, $F\_1$, $F\_2$ be the left and right foci of the ellipse respectively, and $I$ be the incenter of $\triangle PF\_1F\_2$. If $S\_{\triangle IPF\_1}+S\_{\triangle IPF\_2}=2S\_{\triangle IF\_1F\_2}$, determine the eccentricity of the ellipse.

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