
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
Let \(A\), \(B\), and \(C\) be three distinct points on the graph of \(y = (x-1)^2\) such that line \(AB\) is parallel to the \(x\)-axis and \(\triangle ABC\) is a right triangle with area \(2012\). Find the sum of the digits of the \(y\)-coordinate of \(C\).

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