
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 $\mathbf{u}$, $\mathbf{v}$, and $\mathbf{z}$ be unit vectors in $\mathbb{R}^3$. Suppose $\mathbf{w}$ is a vector such that $\mathbf{u} \times (\mathbf{v + z}) + \mathbf{z} = \mathbf{w}$ and $\mathbf{w} \cdot \mathbf{u} = 0$. Compute $\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w})$.

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