
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 that the function $f(x)=ax^{2}+bx+c$ has two zeros at $-2$ and $3$, and $f({-\frac{b}{{2a}}})＞0$, which of the following statements are correct?<br/>①$a \gt 0$;<br/>②The solution set of the inequality $ax+c \gt 0$ is $\{x\left|\right.x \lt 6\}$;<br/>③$a+b+c \gt 0$;<br/>④The solution set of the inequality $cx^{2}-bx+a \lt 0$ is $\left\{{x|{-\frac{1}{3}＜x＜\frac{1}{2}}}\right\}$.

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