
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 there exist vectors $\mathbf{u}$ and $\mathbf{v}$, and a scalar $d$ such that 
\[\mathbf{i} \times ((\mathbf{u} + \mathbf{v}) \times \mathbf{i}) + \mathbf{j} \times ((\mathbf{u} + \mathbf{v}) \times \mathbf{j}) + \mathbf{k} \times ((\mathbf{u} + \mathbf{v}) \times \mathbf{k}) = d (\mathbf{u} + \mathbf{v}).\] 
Determine the value of $d.$ Assume $\mathbf{u}$ is a constant vector and $\mathbf{v}$ can be any vector.

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