
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
The partial correspondence values of the independent variable $x$ and the function value $y$ of the quadratic function $y=ax^{2}+bx+c(a$,$b$,$c$ are constants, $a\neq 0)$ are as follows:<br/><table class="edittable"><tbody><tr><td width="245">$x$</td><td width="40">$\ldots $</td><td width="40" colspan="3">$-2$</td><td width="40" colspan="2">$-1$</td><td width="40" colspan="2">$0$</td><td width="40" colspan="3">$1$</td><td width="40" colspan="3">$2$</td><td width="40" colspan="2">$\ldots $</td></tr><tr><td>$y=ax^{2}+bx+c$</td><td>$\ldots $</td><td colspan="3">$t$</td><td colspan="2">$m$</td><td colspan="2">$-2$</td><td colspan="3">$-2$</td><td colspan="3">$n$</td><td colspan="2">$\ldots $</td></tr></tbody></table>When $x=-\frac{1}{2}$, the corresponding function value $y \gt 0$. The following conclusions are drawn: ①$abc \gt 0$; ②$-2$ and $3$ are the two roots of the equation $ax^{2}+bx+c=0$ with respect to $x$; ③$0 \lt m+n \lt \frac{20}{3}$. The correct conclusions are numbered as ______.

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