
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 $\mathbf{u},$ $\mathbf{v},$ $\mathbf{w}$ be vectors in three-dimensional space, such that $\mathbf{u} + \mathbf{v} + \mathbf{w} = \mathbf{0}$. Calculate the determinant $K$ of the matrix whose column vectors are $\mathbf{u} \times \mathbf{v},$ $\mathbf{v} \times \mathbf{w},$ and $\mathbf{w} \times \mathbf{u}.$ Then, find $K' = K^2$.

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