
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 $f(x)=2\sin(x+\frac{\pi}{3})$.  
$(1)$ If the function $y=f(\omega x-\frac{\pi}{12})$ ($\omega>0$) is decreasing on the interval $({\frac{\pi}{2},\pi})$, find the range of values for $\omega$.  
$(2)$ If the function $g(x)=f(\frac{\pi}{2}-x)$, and the horizontal coordinates of all points on the graph of $g(x)$ are shortened to half of their original length (while the vertical coordinates remain unchanged), and then the graph is shifted to the right by $\frac{\pi}{12}$ units to obtain the graph of $y=h(x)$, if the equation in $x$, $\frac{1}{2}h(x)-k(\sin x+\cos x)=0$, has a solution for $x\in[-\frac{\pi}{12},\frac{5\pi}{12}]$, find the range of real numbers for $k$.

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