
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 coordinates of points $A$, $B$, $C$, and $D$ on the parabola $y=ax^{2}+x+c$ are listed as follows:

| Point | $A$ | $B$ | $C$ | $D$ |
|-------|-----|-----|-----|-----|
| $x$   | $0$ | $1$ | $2$ | $n$ |
| $y$   | $-2$| $1$ | $m$ | $-2$|

$(1)$ Find the values of $a$ and $c$;  
$(2)$ Directly write down the values of $m$ and $n$.

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