
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 four propositions:  
① The negation of the proposition "For all $x \in \mathbb{R}, \cos(x) > 0$" is "There exists an $x \in \mathbb{R}$ such that $\cos(x) \leq 0$".  
② If $0 < a < 1$, then the equation $x^2 + a^x - 3 = 0$ has only one real root.  
③ For any real number $x$, if $f(-x) = f(x)$ and $f'(x) > 0$ when $x > 0$, then $f'(x) < 0$ when $x < 0$.  
④ For a rectangle with area $S$ and perimeter $l$, the ordered pair of real numbers $(6, 8)$ can be a pair of $(S, l)$ that can be obtained. The correct proposition numbers are ____. (Fill in all correct numbers)

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