
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 line $l$ is given by:
\[
\begin{cases}
x=-3+t\cos \alpha \\
y= \sqrt{3}+t\sin \alpha
\end{cases}
\]
where $t$ is a parameter, $0 \leqslant \alpha < \pi$ and $\alpha \neq \frac{\pi}{2}$. With the origin $O$ as the pole and the positive $x$-axis as the polar axis, the polar equation of curve $C$ is $\rho=2\sqrt{3}$. It is known that line $l$ intersects curve $C$ at points $A$ and $B$, and that $|AB|=2\sqrt{3}$.

$(1)$ Find the value of $\alpha$;

$(2)$ Perpendiculars are drawn from $A$ and $B$ to line $l$, meeting the $x$-axis at points $M$ and $N$ respectively. Find $|MN|$.

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