
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
All three vertices of $\bigtriangleup ABC$ lie on the parabola defined by $y = (x-1)^2$. Vertex $A$ is at point $(1,0)$ and $\overline{BC}$ is parallel to the $x$-axis. Given that the area of the triangle is $128$, find the length of $BC$.

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