
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 sets $A=\{x|\frac{1}{2}＜{2}^{x}＜8\}$ and $B=\{x|y=\sqrt{2-\log_{2}(x+2)}\}$.  
$(1)$ Find $\overline{A}∩B$;  
$(2)$ When $x∈(\overline{A}∩B)$, find the range of the function $f\left(x\right)=2^{2-x}$.

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