
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 \( f(x) = x^3 - x^2 \). For a given value of \( c \), the graph of \( f(x) \), together with the graph of the line \( c + x \), split the plane up into regions. Suppose that \( c \) is such that exactly two of these regions have finite area. Find the value of \( c \) that minimizes the sum of the areas of these two regions.

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