
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 parabola $C: y^2 = 2px$ ($p > 0$) that passes through the focus, a line with a slope of $2\sqrt{2}$ intersects the parabola at $A(x_{1}, y_{1})$ and $B(x_{2}, y_{2})$, and $|AB| = \frac{9}{2}$.

(1) Find the equation of this parabola;  
(2) Find a point $D$ on parabola $C$ such that the distance from point $D$ to the line $y = x + 3$ is the shortest.

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