
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
Among the following propositions:
① If $p$ and $q$ are two propositions, then "both $p$ and $q$ are true" is a necessary but not sufficient condition for "either $p$ or $q$ is true";
② If $p$ is: $\exists x \in \mathbb{R}, x^2+2x+2\leq0$, then $\neg p$ is: $\forall x \in \mathbb{R}, x^2+2x+2>0$;
③ If the proposition "$\exists x \in \mathbb{R}, x^2+(a-1)x+1<0$" is false, then the range of real numbers $a$ is $-1\leq a \leq 3$;
④ Given the proposition $p$: $\exists x \in \mathbb{R}, \tan(x)=1$, and the proposition $q$: the solution set of $x^2-3x+2<0$ is $\{x|1<x<2\}$, then the proposition "$\neg p \lor \neg q$" is false. The correct proposition numbers are.

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