
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 that $F\_1$ and $F\_2$ are the left and right foci of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a > 0, b > 0)$, and point $P$ is a point on the right branch of the hyperbola. $M$ is the incenter of $\triangle PF\_1F\_2$, satisfying $S\_{\triangle MPF\_1} = S\_{\triangle MPF\_2} + \lambda S\_{\triangle MF\_1F\_2}$. If the eccentricity of this hyperbola is $3$, then $\lambda = \_\_\_\_\_\_$.
(Note: $S\_{\triangle MPF\_1}$, $S\_{\triangle MPF\_2}$, $S\_{\triangle MF\_1F\_2}$ represent the area of $\triangle MPF\_1$, $\triangle MPF\_2$, $\triangle MF\_1F\_2$ respectively.)

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