
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]


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The following table shows several pairs of values of the independent variable $x$ and the function $y$ of a quadratic function:<br/>

| $x$   | $\ldots $ | $0$ | $1$ | $\frac{3}{2}$ | $2$ | $\ldots $ |
|-------|-----------|-----|-----|---------------|-----|-----------|
| $y$   | $\ldots $ | $-\frac{7}{2}$ | $\frac{1}{2}$ | $1$ | $\frac{1}{2}$ | $\ldots $ |

$(1)$ The coordinates of the intersection point of the quadratic function with the $y$-axis are ______; the axis of symmetry is ______; the coordinates of the vertex are ______.<br/>
$(2)$ Find the analytical expression of the quadratic function.

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