
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 between the independent variable $x$ and the function value $y$ of the quadratic function $y=ax^{2}+bx+c(a$,$b$,$c$ are constants, $a\neq 0)$ is shown in the table below:

| $x$ | $\ldots $ | $-3$ | $-2$ | $-1$ | $0$ | $1$ | $\ldots $ |
|-----|-----------|------|------|------|----|----|-----------|
| $y=ax^{2}+bx+c$ | $\ldots $ | $\frac{5}{2}$ | $4$ | $\frac{9}{2}$ | $4$ | $m$ | $\ldots $ |

Based on the table above, answer the following questions:
$(1)$ Directly write down the values of $c$ and $m$;
$(2)$ Find the analytical expression of this quadratic function.

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