
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 partial corresponding values of $y$ and $x$ of a quadratic function $y=ax^{2}+bx+c$ (where $a$, $b$, and $c$ are constants and $a\neq 0$) are shown in the table below:

| $x$ | $-5$ | $-4$ | $-2$ | $0$ | $2$ |
|-----|------|------|------|----|----|
| $y$ | $6$  | $0$  | $-6$ | $-4$ | $6$ |

Find the roots of the quadratic equation $ax^{2}+bx+c=0$ in terms of $x$.

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