
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 two foci of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{2a} = 1 (a > 0)$, and point $M$ lies on the hyperbola such that $\overrightarrow{MF\_1} \cdot \overrightarrow{MF\_2} = 0$ and $|\overrightarrow{MF\_1}| \cdot |\overrightarrow{MF\_2}| = 4$, find the value of $a$.

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