
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
In the Cartesian coordinate system $xOy$, $F$ is the focus of the parabola $C: y^2 = 2px$ ($p > 0$), and the circle $Q$ passes through points $O$ and $F$, with the distance from the center of circle $Q$ to the directrix of parabola $C$ being $\frac{3}{2}$.  
(1) Find the equation of the parabola $C$;  
(2) Given a point $M(t, 4)$ on the parabola, and two chords $MD$ and $ME$ of the parabola through point $M$ with $MD \perp ME$, determine whether the line $DE$ passes through a fixed point? Explain your reasoning.

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