
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 $U=\mathbb{R}$, let the set $A=\{x|2 < x < 9\}$, and $B=\{x|-2\leqslant x\leqslant 5\}$.  
$(1)$ Find $A\cap B$ and $B\cup(\complement_{U}A)$;  
$(2)$ Given the set $C=\{x|a\leqslant x\leqslant a+2\}$, if $C\subseteq\complement_{U}B$, find the range of the real number $a$.

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