
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 $f(x)=4\sin(x)\cos\left(x+\frac{\pi}{3}\right)+\sqrt{3}$. By changing all the abscissas of the points on the graph of function $f(x)$ to $\frac{2}{3}$ of their original values while keeping the ordinates unchanged, and then shifting the resulting graph to the right by $\frac{\pi}{18}$ units, we obtain the graph of a new function $g(x)$.
- $(1)$ Find the interval of monotonically decreasing values of function $f(x)$ on the interval $[-\frac{\pi}{4}, \frac{\pi}{6}]$.
- $(2)$ If for all $x\in[0,\frac{\pi}{3}]$, the inequality ${g^2}(x)-mg(x)-3\leq 0$ always holds, find the range of real number $m$.

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