
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 $f(x)=2\sin \frac{x}{2}\cos \frac{x}{2}+2\sqrt{3}\cos ^{2}\frac{x}{2}-\sqrt{3}$.  
$(1)$ Find the period of the function $f(x)$.  
$(2)$ If the inequality $|f(x)-m|\leqslant 3$ holds for all $x\in [-\frac{π}{6}$,$\frac{π}{3}]$, find the maximum integer value of $m$.  
$(3)$ Let $g(x)=f(\frac{π}{2}-x)$. The graph of $g(x)$ is obtained by shortening the horizontal coordinates of each point by half (vertical coordinates remain unchanged) and then shifting to the right by $\frac{π}{12}$ units, resulting in the graph of $y=h(x)$. If the equation $\frac{1}{2}h(x)-k(\sin x+\cos x)=0$ has a solution for $x\in [-\frac{π}{12}$,$\frac{5π}{12}]$, find the range of real numbers for $k$.

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