
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 propositions:

1. If proposition $p$: $\forall x \geqslant 0$, $x^{2}+x \geqslant 0$. Then $\neg p$: $\exists x_{0} < 0$, $x_{0}^{2}+x_{0} < 0$

2. If variables $x$, $y$ are linearly related, and their regression equation is $\widehat{y}+x=2$, then $x$, $y$ are positively correlated

3. In $\triangle ABC$, $BC=2$, $AC=3$, $\angle B=\frac{\pi}{3}$, then $\triangle ABC$ is an acute triangle

4. If a wire of length 8 is formed into a rectangular frame, then the probability that the area of the rectangle is greater than 3 is $\frac{1}{2}$

5. Given $a > b > c > 0$, and $2b > a+c$, then $\frac{b}{a-b} > \frac{c}{b-c}$

Among these propositions, the correct ones are ______ (Fill in the serial numbers).

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