
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 sets $A=\{x|x^{2}-6x \lt -5\}$, $B=\{x|1 \lt 2^{x-2}\leqslant 16\}$, and set $C$ as the domain of the function $f\left(x\right)=\lg \left[\left(2a-x\right)\left(x-a-1\right)\right]$, where the universal set is the set of real numbers $R$.
$(1)$ Find $A\cup B$ and $\complement _{R}A$;
$(2)$ If $A\cap C=C$, find the range of real number $a$.

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