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

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