
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 propositions $p: \exists x_0 \in R$ such that $\sin x_0 = \frac{\sqrt{5}}{2}$ and $q: \forall x \in R, x^2 + x + 1 > 0$. Determine the correct conclusion(s) among the following:
$(1)$ Proposition $p$ is true.
$(2)$ Proposition $q$ is false.
$(3)$ The proposition "$\neg p \wedge q$" is true.
$(4)$ The proposition "$p \wedge \neg q$" is false.

The correct option is $\boxed{(3)(4)}$.

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