
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
Given a quadratic function $y=ax^2+bx+c$ ($a\neq0$), the corresponding values of the function $y$ and the independent variable $x$ are shown in the table below:

| $x$ | ... | -2 | -1 | 0 | 1 | 2 | ... |
|-----|-----|----|----|---|---|---|-----|
| $y$ | ... | 5  | 0  | -3| -4| -3| ... |

(1) Find the expression of this quadratic function.  
(2) The function expression corresponding to the graph of this quadratic function being symmetric about the $x$-axis is $y=-(x-1)^2+4$.

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