
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 $F\_1(- \frac{5}{2},0)$ and $F\_2( \frac{5}{2},0)$ are the common foci of the hyperbola $\frac{x^{2}}{4}- \frac{y^{2}}{m}=1$ and the ellipse $\frac{x^{2}}{9}+ \frac{y^{2}}{n}=1$. If point $P$ is one of their intersection points, find the area of $\triangle PF\_1F\_2$.

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