
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
Determine the scalar $p$ such that for any vectors $\mathbf{u},$ $\mathbf{v},$ and $\mathbf{w}$ satisfying $\mathbf{u} + \mathbf{v} + \mathbf{w} = \mathbf{0},$ the equation
\[p (\mathbf{v} \times \mathbf{u}) + 2(\mathbf{v} \times \mathbf{w}) + 3(\mathbf{w} \times \mathbf{u}) = \mathbf{0}\]
holds true.

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