
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
In the middle school third-year mathematics textbook, when using the "plotting method" to draw the graph of the quadratic function $y=ax^{2}+bx+c$, the following table is listed:

| $x$ | $\ldots $ | $-2$ | $-1$ | $0$ | $1$ | $2$ | $\ldots $ |
|-----|-----------|------|------|----|-----|-----|----------|
| $y$ | $\ldots $ | $-6\frac{1}{2}$ | $-4$ | $-2\frac{1}{2}$ | $-2$ | $-2\frac{1}{2}$ | $\ldots $ |

Based on the information in the table, write down the analytical expression of this quadratic function.

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