
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 $A=\{x|a-1 < x < 2a+1\}$ and $B=\{x|0 < x < 1\}$,

$(1)$ If $A \cup B = A$, find the range of the real number $a$;

$(2)$ If $A \cap B = \varnothing$, find the range of the real number $a$.

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