
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 points $A$ and $B$ are the left and right vertices of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a > 0, b > 0)$, the length of the real axis is $4\sqrt{3}$, and the distance from the focus to the asymptote is $\sqrt{3}$.

(1) Find the equation of the hyperbola.

(2) Given that the line $y = \frac{\sqrt{3}}{3}x - 2$ intersects the right branch of the hyperbola at points $M$ and $N$, and there exists a point $D$ on the right branch of the hyperbola such that $\overrightarrow{OM} + \overrightarrow{ON} = t\overrightarrow{OD}$, find the value of $t$ and the coordinates of point $D$.

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