
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 the function $f\left( x \right)=2\sqrt{3}{{\sin }^{2}}\left( \frac{\pi }{4}+x \right)+2\sin \left( \frac{\pi }{4}+x \right)\cos \left( \frac{\pi }{4}+x \right)$

(I) Find the monotonically increasing interval and its center of symmetry for the function $f\left( x \right)$;

(II) In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and angle $A$ satisfies $f\left( A \right)=\sqrt{3}+1$. If $a=3$, and the length of the median on edge $BC$ is $3$, find the area $S$ of $\triangle ABC$.

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