
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 $f(x)=A\sin (\omega x+\varphi )+h$ $(A > 0,\omega > 0,|\varphi| < \pi)$. Within one period, when $x= \frac{\pi}{12}$, $y$ reaches its maximum value of $6$, and when $x= \frac{7\pi}{12}$, $y$ reaches its minimum value of $0$.

$(1)$ Find the expression for the function $f(x)$;

$(2)$ Find the intervals of monotonic increase and the coordinates of the symmetry center for the function $f(x)$;

$(3)$ When $x\in \left[ -\frac{\pi }{12},\frac{\pi }{6} \right]$, the graph of the function $y=mf(x)-1$ intersects with the $x$-axis, find the range of the real number $m$.

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