
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 hyperbola's real axis length 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 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 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}
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