
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 triangle $ABC$ with internal angles $A$, $B$, and $C$, and vectors $\overrightarrow{m}=(\sqrt{3}\sin A, \sin B)$ and $\overrightarrow{n}=(\cos B, \sqrt{3}\cos A)$, such that $\overrightarrow{m}\cdot\overrightarrow{n}=1+\cos(A+B)$, determine the value of angle $C$.

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