
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 table below gives some corresponding values of $y$ and $x$ in the function $y=x^{2}+bx+c$.

| $x$ | $\ldots $ | $1$ | $2$ | $3$ | $4$ | $5$ | $\ldots $ |
|-----|-----------|-----|-----|-----|-----|-----|-----------|
| $y$ | $\ldots $ | $0$ | $m$ | $-4$| $n$ | $0$ | $\ldots $ |

$(1)$ The parabola $y=x^{2}+bx+c$ opens ______, with the axis of symmetry being ______;  
$(2)$ When $y \gt 0$, the range of values for $x$ is ______;  
$(3)$ Find the analytical expression of the function $y=x^{2}+bx+c$, and express it in vertex form as ______.

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