
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 sets $A=\{x|y=\sqrt{x-2}+{(x-3)}^{0}\}$, $B=\{x|0\leqslant x-1\leqslant 4\}$, $C=\{x|m-1 < x < 4m, m\in \mathbb{R}\}$.

$(1)$ Find the sets $A\cap B, A\cup B$;

$(2)$ If $B\subseteq C$, find the range of the real number $m$.

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