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

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

$(1)$ Find the analytical expression of this parabola;<br/>$(2)$ The coordinates of the vertex of the parabola are _______ and as $x \gt 1$, $y$ increases as $x$ _______ (fill in "increases" or "decreases");<br/>$(3)$ Write down directly the range of values of $y$ when $-1 \lt x \lt 2$.

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