
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 the eccentricity of the hyperbola $C$ is $\sqrt{3}$, and it passes through the point $(\sqrt{3},0)$, intersects the hyperbola $C$ at points $A$ and $B$ with a line $l$ passing through the right focus $F_{2}$ of the hyperbola at an angle of $\frac{π}{3}$, where $O$ is the origin and $F_{1}$ is the left focus.  
$(1)$ Write down the equation of the line $l$;  
$(2)$ Find the standard equation of the hyperbola;  
$(3)$ Find the area of $\triangle AOB$.

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