
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 conclusions, the correct proposition numbers are __________. (Write down all the correct proposition numbers.)

① The value of the integral $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos x \, dx$ is $2$;

② For the proposition $p$: there exists $x_0 \in \mathbb{R}$, such that $x_0^2 + x_0 + 1 < 0$, then $\neg p$: for any $x \in \mathbb{R}$, $x^2 + x + 1 > 0$;

③ If a function $f(x)$ defined on $\mathbb{R}$ satisfies $f(x+2) = -f(x)$, then $4$ is a period of the function $f(x)$;

④ In the sequence $\{a_n\}$, where $S_n = 3^n + a$, $\{a_n\}$ is a geometric sequence if and only if $a = -1$.

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