
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 $\mathbf{a}, \mathbf{b}, \mathbf{c},$ and $\mathbf{d}$ be four mutually orthogonal unit vectors, such that
\[\mathbf{a} = p (\mathbf{a} \times \mathbf{b}) + q (\mathbf{b} \times \mathbf{c}) + r (\mathbf{c} \times \mathbf{a}) + s (\mathbf{d} \times \mathbf{a})\] for some scalars $p,$ $q,$ $r,$ and $s,$ and $\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = 1.$ Find $p + q + r + s.$

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