
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 that the vertex of parabola $C$ is at the origin, and $F\left(\frac{1}{2}, 0\right)$ is the focus of the parabola.  
(1) Find the equation of parabola $C$;  
(2) A line $l$ passing through point $F$ intersects the moving parabola $C$ at points $A$ and $B$, and intersects the circle $M: \left(x- \frac{3}{2}\right)^2 + (y-8)^2 = 49$ at points $D$ and $E$, with $D$ and $E$ located on the line segment $AB$. If $|AD| = |BE|$, find the equation of line $l$.

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