
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 $A=\{x|1\leq x<6\}$, $B=\{x|2<x<9\}$.  
(1) Find separately: $A\cap B$, $A\cup (\complement_{\mathbb{R}} B)$;  
(2) Given $C=\{x|a<x<a+1\}$, if $C\subseteq B$, find the range of the real number $a$.

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