
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 $f(x)=2 \sqrt {3}\sin ^{2}( \frac {π}{4}+x)+2\sin ( \frac {π}{4}+x)\cos ( \frac {π}{4}+x)$.
(I) Find the monotonically increasing interval and its symmetry center of the function $f(x)$;
(II) In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. If angle $A$ satisfies $f(A)= \sqrt {3}+1$, and $a=3$, 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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