
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 point P (3, 4) and a circle C: $(x-2)^2+y^2=4$, let A and B be two moving points on circle C such that $|AB|=2\sqrt{3}$. Determine the range of values for the dot product $\vec{OP}\cdot(\vec{OA}+\vec{OB})$ (where O is the origin of the coordinate system).

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