
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 $C$: $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1(a>0,b>0)$ with one vertex on the imaginary axis denoted as $D$, and $F_{1}$ and $F_{2}$ as the left and right foci of $C$ respectively. The line $x=2a$ intersects $C$ at points $A$ and $B$. If the centroid of triangle $ABD$ lies on a circle with diameter $F_{1}F_{2}$, find the equation of the asymptote of $C$.

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