
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 following statements: 
1. The negation of the proposition "There exists an $x \in \mathbb{R}$ such that $x^2 - 2x - 3 > 0$" is "For all $x \in \mathbb{R}$, $x^2 - 2x - 3 < 0$".
2. If the proposition "not $p$" is true, and the proposition "$p$ or $q$" is true, then the proposition $q$ is true.
3. If $q$ is a necessary but not sufficient condition for $p$, then the negation of the proposition "if $p$ then $q$" is true, and the contrapositive is false.
Indicate which of these propositions are correct.

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