
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 functions $f(x)=2x^{2}-3$ and $g(x)=k\sin\left(\frac{{\pi x}}{4}-\frac{\pi}{3}\right)$.  
$(1)$ If for any $x\in\left[-\frac{\pi}{2},\frac{{2\pi}}{3}\right]$, $f\left(\cos x\right)\leq a\cos x+1$, find the range of $a$.  
$(2)$ If for any $x_1\in\left[-2,\sqrt{3}\right]$, there exists $x_{2}\in \left(0,4\right)$ such that $g(x_{2})=f(x_{1})$, find the range of $k$.

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