
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 rectangular coordinate system $(xOy)$, the parametric equations of curve $C$ are given by $\begin{cases} x=3+\sqrt{5}\cos\theta \\ y=\sqrt{5}\sin\theta \end{cases}$ (where $\theta\in\mathbb{R}$).
(1) Establish the polar coordinate system with the coordinate origin as the pole and the non-negative $x$-axis as the polar axis, and derive the polar equation of curve $C$.
(2) The parametric equations of line $l$ are given by $\begin{cases} x=1+t\cos\alpha \\ y=t\sin\alpha \end{cases}$ (where $t\in\mathbb{R}$ and $\alpha$ is a constant). Line $l$ intersects curve $C$ at points $A$ and $B$, with $|AB|=2\sqrt{3}$. Determine the slope of line $l$.

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