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

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