
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 the hyperbola $C$: $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ ($a > b > 0$) with its right focus at $F_2$, and point $A(2,1)$ is on the hyperbola $C$. Point $B$ is symmetric to point $A$ with respect to the origin. If $\angle AF_2B=\dfrac{2\pi}{3}$ and the area of $\triangle ABF_2=\sqrt{3}$, then the standard equation of the hyperbola $C$ is \_\_\_\_\_\_\_\_.

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