
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 $\mathbb{R}$, $A=\{x|2\leqslant x < 4\}$, $B=\{x|3x-7\geqslant 8-2x\}$.  
$(1)$ Find $A\cup(\complement_{\mathbb{R}}B)$.  
$(2)$ If $C=\{x|a-1\leqslant x\leqslant a+3\}$ and $A\cap C=A$, find the range of the real number $a$.

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