
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 set $A=\{x|-1\leqslant x\leqslant 10\}$ and the set $B=\{x|2x-6\geqslant 0\}$. Find the complement of $A\cup B$ in the set of real numbers, denoted as $∁\_R(A\cup B)$; Also given the set $C=\{x|a < x < a+1\}$, and $C\subseteq A$, find the range of values for the real number $a$.

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