
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 function $f(x) = \sqrt{x-5} - \frac{1}{\sqrt{8-x}}$, its domain is the set $A$, $B = \{ x \in Z | 3 < x < 11 \}$, $C = \{ x \in R | x < a \text{ or } x > a+1 \}$.  
$(1)$ Find $A$, $(\complement_{R}A) \cap B$;  
$(2)$ If $A \cup C = R$, find the range of the real number $a$.

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