
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 $f(x) = \sqrt{x-3} - \frac{1}{\sqrt{7-x}}$, its domain is set $A$, $B=\{x|2<x<10\}$, $C=\{x|a<x<2a+1\}$.  
(1) Find $A\cup B$, $(\complement_R A) \cap B$  
(2) If $B\cup C=B$, find the range of the real number $a$.

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