
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 $l: ax+y+b=0$ intersects with the circle $O: x^{2}+y^{2}=4$ at points $A$ and $B$, and $M(\sqrt{3},-1)$, and $\overrightarrow{OA}+ \overrightarrow{OB}= \frac{2}{3} \overrightarrow{OM}$, calculate the value of $\sqrt{3}ab$.

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