
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 distinct unit vectors in space such that:
\[\mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{c} = \mathbf{c} \cdot \mathbf{d} = -\frac{1}{7}\]
and
\[\mathbf{b} \cdot \mathbf{d} = \frac{1}{3}.\]
Find the value of $\mathbf{a} \cdot \mathbf{d}$.

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