
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. The perpendiculars to line $l$ are drawn through points A and B, intersecting the x-axis at points C and D, respectively. If $AB = 2\sqrt{3}$, find the value of $|CD|$.

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