
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 universal set $U = R$, set $A=\{x||2x-a|\geqslant 2\}$, $B=\{x|2 \lt 2^{x} \lt 8\}$.  
$(1)$ When $a=2$, find $A\cap B$;  
$(2)$ In the case of $A\cap B\neq \varnothing $ selected as a condition, find the range of real number $a$ satisfying $B\subseteq A$.

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