
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 $\mathbf{i}, \mathbf{j}, \mathbf{k}, \mathbf{w}$ be orthogonal unit vectors in a four-dimensional space. For all vectors $\mathbf{v}$ in this space, determine the scalar $c$ such that
\[\mathbf{i} \times (\mathbf{v} \times \mathbf{i}) + \mathbf{j} \times (\mathbf{v} \times \mathbf{j}) + \mathbf{k} \times (\mathbf{v} \times \mathbf{k}) + \mathbf{w} \times (\mathbf{v} \times \mathbf{w}) = c \mathbf{v}.\]

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