
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
Which of the following statements is correct?

(1) $\{x|x^2-x+1=0, x\in \mathbb{R}\}$ is an empty set.  
(2) Any set must have two subsets.  
(3) The solution to $x^2+1=2x$ can be represented as $\{1, 1\}$.  
(4) The set $\{y|y=x^2-1\}$ and the set $\{(x,y)|y=x^2-1\}$ are the same set.

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