
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
For any vector $\mathbf{v}$ in $\mathbb{R}^3$, consider a new operation where
\[
\mathbf{i} \times (\mathbf{v} \times (\mathbf{i}+\mathbf{j})) + \mathbf{j} \times (\mathbf{v} \times (\mathbf{j}+\mathbf{k})) + \mathbf{k} \times (\mathbf{v} \times (\mathbf{k}+\mathbf{i})) = d \mathbf{v}.
\]
Determine the scalar $d$.

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