
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 the line $l$: $mx+y+3m- \sqrt {3}=0$ intersects the circle $x^{2}+y^{2}=12$ at points $A$ and $B$. Perpendiculars to $l$ through $A$ and $B$ intersect the $x$-axis at points $C$ and $D$ respectively. If $|AB|=2 \sqrt {3}$, then $|CD|=$ \_\_\_\_\_\_.

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