
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 a Cartesian coordinate system $xOy$, the parametric equation of line $l$ is 
$$
\begin{cases} 
x=-3+t\cos \alpha \\ 
y= \sqrt{3}+t\sin \alpha
\end{cases}
(t \text{ is the parameter}, 0\leqslant \alpha < \pi \text{ and } \alpha \neq \frac {\pi}{2}),
$$ 
and a polar coordinate system with 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 with curve $C$ at points $A$ and $B$, and the length of $AB$ is $2\sqrt{3}$.

(1) Determine the value of $\alpha$;  
(2) Construct perpendicular lines from points $A$ and $B$ to the $x$-axis, intersecting at points $M$ and $N$, respectively. Find the length of $|MN|$.

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