
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 corresponding values of the abscissa $x$ and ordinate $y$ of some points on the graph of the quadratic function $y=ax^{2}+bx+c\left(a\neq 0\right)$ are shown in the table below:

| $x$ | $\ldots $ | $-3$ | $-2$ | $-1$ | $0$ | $1$ | $2$ | $n$ | $4$ | $\ldots $ |
|-----|-----------|------|------|------|----|----|----|----|----|-----------|
| $y$ | $\ldots $ | $15$ | $m$  | $3$  | $0$| $-1$| $0$ | $3$ | $8$| $\ldots $ |

$(1)$ The axis of symmetry of the graph of this quadratic function is the line ______;  
$(2)$ $m=$______, $n=$______;  
$(3)$ According to the information in the table, the solutions of the equation $ax^{2}+bx+c=0\left(a\neq 0\right)$ are ______.

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