
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
Determine whether the following propositions are universal or existential propositions, and rewrite these propositions using mathematical quantifier symbols:  
(1) For any $m>1$, the equation $x^2-2x+m=0$ has no real roots;  
(2) There exists a pair of real numbers $x, y$ such that $2x+3y+3>0$;  
(3) There exists a triangle that does not have a circumcircle;  
(4) The square of a real number is greater than or equal to 0.

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