
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 $f(x)= \sqrt {\log _{2}(x-1)}$ with domain set $A$, and function $g(x)=( \frac {1}{2})^{x}(-1\leqslant x\leqslant 0)$ with range set $B$.
  1. Find $A∩B$;
  2. If set $C=[a,2a-1]$, and $C∪B=B$, find the range of real number $a$.

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