
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 the hyperbola $C\_1$: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a > b > 0)$ with left and right foci $F\_1$ and $F\_2$, respectively, and hyperbola $C\_2$: $\frac{x^2}{16} - \frac{y^2}{4} = 1$, determine the length of the major axis of hyperbola $C\_1$ given that point $M$ lies on one of the asymptotes of hyperbola $C\_1$, $OM \perp MF\_2$, and the area of $\triangle OMF\_2$ is $16$.

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