
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
Let the set $A=\left\{ x\mid 2a < x < a+1 \right\}$, and the set $B=\left\{ x\mid x < -1 \text{ or } x > 3\right\}$.

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

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

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