
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 values of $y$ and $x$ for the quadratic function $y=ax^{2}+bx+c$ as shown in the table:

| $x$ | $\ldots $ | $-2$ | $-1$ | $1$ | $2$ | $4$ | $\ldots $ |
|-----|-----------|------|------|-----|-----|-----|-----------|
| $y$ | $\ldots $ | $21$ | $12$ | $0$ | $-3$ | $-3$ | $\ldots $ |

Determine the solutions of the quadratic equation $ax^{2}+bx+c=0$.

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