
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 set $A=\{x|1 \lt x \lt 3\}$, set $B=\{x|2m \lt x \lt 1-m\}$.  
$(1)$ If "$x\in A$" is a sufficient condition for "$x\in B$", find the range of real number $m$;  
$(2)$ If $A\cap B=\varnothing$, find the range of real number $m$.

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