
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
In the Cartesian coordinate system $xoy$, the curve $C$ is given by the parametric equations
\[
\begin{cases}
x= \sqrt{5}\cos \theta \\
y=3+ \sqrt{5}\sin \theta
\end{cases} 
\]
where $\theta$ is the parameter.

$(1)$ Establish a polar coordinate system with the origin $O$ as the pole and the positive direction of the $x$-axis as the polar axis. Find the polar equation of curve $C$.

$(2)$ The parametric equations of line $l$ are
\[
\begin{cases}
x=t\cos \alpha \\
y=t\sin \alpha
\end{cases}
\]
where $\alpha$ is a parameter. Line $l$ intersects curve $C$ at points $A$ and $B$, and the distance between $A$ and $B$ is $2\sqrt{3}$. Find the slope of line $l$.

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