
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 $f(x)=2\sin(\omega x+\varphi)$, where $\omega\gt 0$ and $|\varphi|\lt \pi$, the horizontal distance between the adjacent highest and lowest points on the graph of $f(x)$ is $\frac{\pi}{2}$; _______;  
$(1)$① $f(x)$ has a symmetry axis $x=-\frac{\pi}{3}$ and $f(\frac{\pi}{6})>f(1)$;  
② $f(x)$ has a symmetry center $({\frac{5\pi}{12}},0)$ and is monotonically decreasing on $[{\frac{\pi}{6},\frac{2\pi}{3}}]$;  
③ $f(x)$ is shifted to the left by $\frac{\pi}{6}$ units to obtain a graph that is symmetric about the $y$-axis and $f(0)>0$.  
Choose one of the above three conditions to fill in the blank line above, and then determine the analytical expression of the function;  
$(2)$ In case (1), let $h(x)=\frac{1}{2}f(x)-\cos(2x)$, $g(x)=h[h(x)]$. If there exists $x\in[{\frac{\pi}{12},\frac{\pi}{3}}]$ such that $g^{2}(x)+(2-a)g(x)+3-a\leqslant 0$ holds, find the range of real number $a$.

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