
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 $F$ is the right focus of the hyperbola $C$: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a > 0, b > 0)$, and $O$ is the center of the hyperbola $C$. The line $y = \sqrt{m}x$ is an asymptote of the hyperbola $C$. Construct an equilateral triangle $AOF$ with the segment $OF$ as one side. If point $A$ lies on the hyperbola $C$, calculate the value of $m$.

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