
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 $\mathbf{u}_1 = \mathbf{i} + \mathbf{j}$, $\mathbf{u}_2 = \mathbf{j} + \mathbf{k}$, and $\mathbf{u}_3 = \mathbf{k} + \mathbf{i}$. For all vectors $\mathbf{v}$, find a scalar $d$ such that
\[\mathbf{u}_1 \times (\mathbf{v} \times \mathbf{u}_1) + \mathbf{u}_2 \times (\mathbf{v} \times \mathbf{u}_2) + \mathbf{u}_3 \times (\mathbf{v} \times \mathbf{u}_3) = d \mathbf{v}.\]

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