
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 parametric equation of the curve \(C\) is
\[
\begin{cases}
    x = 3 + \sqrt{5}\cos \theta, \\
    y = \sqrt{5}\sin \theta
\end{cases}
\]
(where parameter \(\theta \in \mathbb{R}\)).
(1) Establish a polar coordinate system with the origin as the pole and the non-negative half of the \(x\)-axis as the polar axis, and find the polar equation of the curve \(C\).
(2) The parametric equation of the line \(l\) is
\[
\begin{cases}
    x = 1 + t\cos \alpha, \\
    y = t\sin \alpha
\end{cases}
\]
(where parameter \(t \in \mathbb{R}\) and \(\alpha\) is a constant), line \(l\) intersects curve \(C\) at points \(A\) and \(B\), and the distance \(|AB| = 2\sqrt{3}\). Find the slope of the line \(l\).

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