
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 $U=\mathbb{R}$, $A=\{x||x-1|\geqslant 1\}$, $B$ is the domain of the function $f(x)= \sqrt {2- \frac {x+3}{x+1}}$, and $C$ is the domain of $g(x)=\lg [(x-a-1)(2a-x)](a < 1)$;
$(1)$ Find $A∩B$ and $C_{U}(A∪B)$;
$(2)$ If $C⊆B$, find the range of values for the real number $a$.

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