
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 that $y$ is a quadratic function of $x$, and $x$ and $y$ satisfy the following table:

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

$(1)$ Find the analytical expression of the quadratic function;  
$(2)$ When $0 \lt x \lt 3$, directly write down the range of values for $y$.

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