
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 the line $m: y = 2x - 16$, and the parabola $C: y^2 = ax$ ($a > 0$).
(1) When the focus of parabola $C$ is on line $m$, determine the equation of parabola $C$;
(2) If the three vertices of $\triangle ABC$ are all on the parabola $C$ determined in (1), and the ordinate of point $A$ is $y = 8$, and the centroid of $\triangle ABC$ is exactly at the focus of parabola $C$, find the equation of line $BC$.

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