
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 $A=\left\{{x|{2-\frac{x+3}{x+1}≥0}}\right\}$ and $B=\{x\left|\right.\left(x-a-1\right)\left(x-2a\right) \lt 0\}$, where $a \lt 1$. Find:<br/>
$(1)$ The sets $A$ and $B$;<br/>
$(2)$ If $A\cup B=A$, find the range of real numbers for $a$.

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