
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 a hyperbola $M: x^2 - \frac{y^2}{m} = 1 (m > 0)$ with an eccentricity of $2$, let $P$ be a point on the hyperbola. If the sum of the distances from $P$ to the left and right foci, $F\_1$ and $F\_2$, is $10$, then what is the value of $\overrightarrow{PF\_1} \cdot \overrightarrow{PF\_2}$?

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