
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 parametric equations of curve $C$ are $\begin{cases} x=2+\sqrt{5}\cos\alpha, \\ y=1+\sqrt{5}\sin\alpha \end{cases}$ ($\alpha$ is the parameter), with the origin $O$ of the Cartesian coordinate system as the pole and the positive half-axis of $x$ as the polar axis to establish the polar coordinate system.

$(1)$ Find the polar equation of curve $C$;

$(2)$ Let $l_{1}: \theta =\frac{\pi }{6}$, $l_{2}: \theta =\frac{\pi }{3}$. If $l_{1}$, $l_{2}$ intersect with curve $C$ at two points $A$, $B$ different from the origin, find the area of $\triangle AOB$.

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