
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 sets $$A=\{x|y= \sqrt {3^{x}-3}\}$$, and $$B=\{x|y= \sqrt {\log_{0.5}x+1}\}$$.
(1) Find $A\cap B$;
(2) Let the set $C=\{x|a-2\leq x\leq 2a-1\}$, and $C\cup (A\cap B)=C$, find the range of the real number $a$.

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