
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 that line segment $AB$ is a moving chord on circle $C: (x-1)^2 + (y-1)^2 = 4$, and $|AB| = 2\sqrt{3}$, let point $O$ be the origin. Find the maximum value of $|\overrightarrow{OA} + \overrightarrow{OB}|$ which is ______; if the lines $l_{1}: x-my-3m+1=0$ and $l_{2}: mx+y+3m+1=0$ intersect at point $M$, find the minimum value of $\overrightarrow{MA} \cdot \overrightarrow{MB}$ which is ______.

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