
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
Let the universal set be $U=\mathbb{R}$, and the sets be $A=\{x|(x+3)(4-x)\leqslant 0\}$, $B=\{x|\log_{2}(x+2) < 3\}$.  
$(1)$ Find $A\cap \complement_{U}B$;  
$(2)$ Given $C=\{x|2a < x < a+1\}$, if $C\subseteq A\cup B$, find the range of the real number $a$.

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