
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
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and $c^{2}=a^{2}+b^{2}-ab $.

(I) If $\tan A-\tan B= \frac{ \sqrt{3}}{2}\left(1+\tan A\cdot\tan B\right) $, find angle $B$;

(II) Let $\overset{⇀}{m}=\left(\sin A,1\right), \overset{⇀}{n}=\left(3,\cos 2A\right) $, try to find the maximum value of $\overset{⇀}{m}\cdot \overset{⇀}{n} $.

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