
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
If $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}$ are vectors such that $\|\mathbf{a}\| = \|\mathbf{b}\| = 2,$ $\|\mathbf{a} - \mathbf{b}\| = 2\sqrt{2},$ and
\[\mathbf{c} - 2\mathbf{a} - \mathbf{b} = 2 (\mathbf{a} \times \mathbf{b}),\]find $\mathbf{a} \cdot \mathbf{c}.$

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