
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 with the equation $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 (a,b > 0)$, its transverse axis length is $4\sqrt{3}$, and the distance between focus and asymptote is $\sqrt{3}$.

1. Determine the equation of this hyperbola.
2. It is known that the line $y=\frac{\sqrt{3}}{3}x - 2$ intersects the right branch of the hyperbola at points $A$ and $B$. Additionally, there exists a point $C$ on the right branch of the hyperbola such that $\overrightarrow{OM} + \overrightarrow{OB} = m\overrightarrow{OC}$. Find the value of $m$ and the coordinates of point $C$.

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