
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
State the negation $\neg p$ of the following propositions $p$, and determine the truth value of the proposition $\neg p$:

(1) $p: \forall x \in \mathbb{R}, x^2 + x + 1 > 0$;                   

(2) $p: \exists x_0, y_0 \in \mathbb{R}, \sqrt{x_0 - 1} + (y_0 + 1)^2 = 0$

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