
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 $f(x)=4\sin\frac{{ωx}}{2}\cos\frac{{ωx}}{2}+1$, where the constant $\omega\ \ \gt 0$.<br/>$(1)$ If $y=f\left(x\right)$ is monotonically increasing on $[{-\frac{π}{4},\frac{{3π}}{4}}]$, find the range of $\omega$;<br/>$(2)$ If $\omega\ \ \lt 4$, the graph of the function $y=f\left(x\right)$ is shifted to the left by $\frac{π}{3}$ units, resulting in the graph of the function $y=g\left(x\right)$ passing through $P({\frac{π}{6},1})$. If the function $g\left(x\right)$ has at least $30$ zeros in the interval $\left[a,b\right](a$,$b\in R$ and $a \lt b)$, find the minimum value of $b-a$ in the intervals that satisfy the above conditions;<br/>$(3)$ Under the conditions of $(2)$, if the inequality $g^{2}(x)-mg\left(x\right)-1\leqslant 0$ holds for any $x∈[{-\frac{π}{6},\frac{π}{{12}}}]$, find the range of real number $m$.

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