
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 circle $C$ with the equation $(x-3)^2+y^2=4$ in the Cartesian coordinate system $xOy$, and points $A$ and $B$ on circle $C$ such that $|AB|=2\sqrt{3}$, find the maximum value of $|\overrightarrow{OA}+\overrightarrow{OB}|$.

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