
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 $U=\mathbb{R}$, the set $A=\{x | x^2 - 4x \leq 0\}$, and $B=\{x | m \leq x \leq m+2\}$.
$(1)$ If $m=3$, determine $\complement_UB$ and $A \cup B$;
$(2)$ If $B \subseteq A$, find the range of the real number $m$;
$(3)$ If $A \cap B = \emptyset$, find the range of the real number $m$.

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