
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 \(\mathbf{u}, \mathbf{v}, \mathbf{w},\) and \(\mathbf{z}\) be vectors such that \(\|\mathbf{u}\| = 2,\) \(\|\mathbf{v}\| = 6,\) \(\|\mathbf{w}\| = 3,\) and \(\|\mathbf{z}\| = 1,\) and
\[\mathbf{u} + \mathbf{v} + \mathbf{w} + \mathbf{z} = \mathbf{0}.\] Compute \(\mathbf{u} \cdot \mathbf{v} + \mathbf{u} \cdot \mathbf{w} + \mathbf{u} \cdot \mathbf{z} + \mathbf{v} \cdot \mathbf{w} + \mathbf{v} \cdot \mathbf{z} + \mathbf{w} \cdot \mathbf{z}.\)

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