
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 $f(x)= \sqrt{3}\sin \left(\omega x- \frac{\pi}{6}\right)+b$ $(\omega > 0)$, and the minimum distance from the center of symmetry of the graph of the function to the axis of symmetry is $\frac{\pi}{4}$. When $x\in\left[ \frac{5\pi}{12}, \frac{7\pi}{6}\right]$ , the maximum value of $f(x)$ is $1$.

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

$(2)$ The graph of the function $f(x)$ is translated to the right by $\frac{\pi}{12}$ units to obtain the graph of the function $g(x)$. If $g(x)-3\leqslant m\leqslant g(x)+3$ holds for all $x\in\left[0, \frac{\pi}{3}\right]$, find the range of the real number $m$.

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