
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 $U = \mathbb{R}$, set $P=\{x|x(x-2)\geqslant 0\}$, $M=\{x|a \lt x \lt a+3\}$.  
$(Ⅰ)$ Find the complement of set $P$ with respect to $U$.  
$(Ⅱ)$ If $a=1$, find the intersection of sets $P$ and $M$.  
$(Ⅲ)$ If $\complement _{U}P\subseteq M$, find the range of real number $a$.

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