
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
1. Proposition p: There exists an $x \in \mathbb{Z}$, such that $x^2 - 2x - 3 = 0$. The negation of p is:  
2. The negation of the proposition "For all $x \in \mathbb{R}$, then $x^2 + 3 \geq 2x$" is:  
3. Proposition q: If $x > 1$ and $y > 1$, then $x + y > 2$. The negation of this proposition is:

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