
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 $\mathbf{w}$ be vectors such that $\|\mathbf{u}\| = \|\mathbf{v}\| = 2$, $\|\mathbf{u} - \mathbf{v}\| = 2\sqrt{2}$, and
\[\mathbf{w} - 2\mathbf{u} + \mathbf{v} = 2 (\mathbf{u} \times \mathbf{v}).\]
Find $\mathbf{v} \cdot \mathbf{w}$.

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