
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 \(\mathbf{w}\), \(\mathbf{x}\), \(\mathbf{y}\), and \(\mathbf{z}\) be four distinct unit vectors in space such that
\[
\mathbf{w} \cdot \mathbf{x} = 0, \quad \mathbf{w} \cdot \mathbf{y} = \frac{1}{5}, \quad \mathbf{x} \cdot \mathbf{y} = -\frac{1}{5}, \quad \mathbf{x} \cdot \mathbf{z} = \frac{1}{7}, \quad \mathbf{y} \cdot \mathbf{z} = -\frac{2}{7}.
\]
Find \(\mathbf{w} \cdot \mathbf{z}\).

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