
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 $f(x)=2\sin \frac{3}{2}x \cdot (\cos \frac{3}{2}x+\sqrt{3}\sin \frac{3}{2}x)-\sqrt{3}$.

(1) Find the analytic expression of the function $f(x)$;

(2) Find the interval where the function $f(x)$ is monotonically increasing;

(3) When $x\in [0,\frac{\pi }{6}]$, the inequality $mf\left( x \right)+2m\geqslant f\left( x \right)$ always holds. Find the range of values for the real number $m$.

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