
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 set $A=\{x\in \mathbb{R} | mx^2-2x+1=0\}$, find the range of the real number $m$ under the following conditions:  
(Ⅰ) $A=\emptyset$;  
(Ⅱ) $A$ has exactly two subsets;  
(Ⅲ) $A\cap\left(\frac{1}{2}, 2\right) \neq \emptyset$

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