
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 the origin $O$ and the point $F$. The distance from the center of the circle $Q$ to the axis of symmetry of the parabola $C$ is $\frac{3}{2}$.
(1) Find the equation of the parabola $C$;
(2) Draw a line $L$ through $F$ with an inclination angle of $60^{\circ}$, intersecting the parabola $C$ at points $A$ and $B$. Find the area of the triangle $\triangle OAB$;
(3) Given a point $M(4,4)$ on the parabola, draw two chords $MD$ and $ME$ passing through $M$, such that $MD \perp ME$. 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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