
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 $f(x)=\sin ^{2}x+ \frac{ \sqrt{3}}{2}\sin 2x$.
(1) Find the interval(s) where the function $f(x)$ is monotonically decreasing.
(2) In $\triangle ABC$, if $f(\frac{A}{2})=1$ and the area of the triangle is $3\sqrt{3}$, find the minimum value of side $a$.

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