
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 vectors $\mathbf{u}, \mathbf{v}, \mathbf{w}$, and $\mathbf{x}$ such that $\mathbf{u} + \mathbf{v} + \mathbf{w} + \mathbf{x} = \mathbf{0}$, determine the scalar $m$ such that the equation
\[m (\mathbf{v} \times \mathbf{u}) + (\mathbf{v} \times \mathbf{w}) + (\mathbf{w} \times \mathbf{x}) + (\mathbf{x} \times \mathbf{u}) = \mathbf{0}\]
holds true for any set of such vectors.

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