
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 $f(x)=A\sin (\omega x+\varphi)$ ($A > 0$, $\omega > 0$, $|\varphi| < \pi$), within the same period, when $x=\frac{\pi}{12}$, $f(x)$ reaches its maximum value of $3$; when $x=\frac{7\pi}{12}$, $f(x)$ reaches its minimum value of $-3$.

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

$(2)$ If $x\in \left[ -\frac{\pi}{3},\frac{\pi}{6} \right]$ and the function $h(x)=2f(x)+1-m$ has two zeros, find the range of the real number $m$.

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