
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 parabola \(y = x^2\), points \(A\), \(B\), and \(C\) form an isosceles right triangle with point \(B\) being the right-angle vertex.
1. Find the coordinates of the vertex \(B\) if the slope of the line segment \(BC\) is \(k\).
2. When the slope \(k\) of the line \(BC\) changes, find the minimum area of the isosceles right triangle \(ABC\).

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