
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 $\mathbf{a}, \mathbf{b},$ and $\mathbf{c}$ be vectors such that $\|\mathbf{a}\| = \|\mathbf{b}\| = 1$, $\|\mathbf{a} + \mathbf{b}\| = \sqrt{2}$, and
\[
\mathbf{c} - 2\mathbf{a} - \mathbf{b} = 2 (\mathbf{a} \times \mathbf{b}).
\]
Find the value of $\mathbf{b} \cdot \mathbf{c}$.

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