
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
Several propositions are listed as follows:
1. The negation of the proposition "There exists an $x \in \mathbb{R}$ such that $\log_a(x^2+1) > 3$" is "For all $x \in \mathbb{R}$, $x^2 + 1 < 3$";
2. Given two simple propositions $p$ and $q$, if the proposition "$p \lor q$" is false, then the proposition "$\neg p \land \neg q$" is true;
3. "$a > 2$" is a sufficient but not necessary condition for "$a > 5$";
4. If the function $f(x) = (x + 1)(x + a)$ is an even function, then $a = -1$.

Identify the correct propositions.

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