
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
A quadratic function $f(x)$ intersects the $x$-axis at two points $(-2,0)$ and $(4,0)$, with its vertex located at $(1, -\frac{9}{2})$.
(1) Find the analytic expression of $f(x)$;
(2) Identify the direction of the parabolic opening, the axis of symmetry, and the vertex coordinates;
(3) Analyze the monotonicity of the function and find its maximum or minimum value.

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