
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 three orthogonal unit vectors, with an angle of $\frac{\pi}{4}$ between $\mathbf{a}$ and $\mathbf{b}$, compute the volume of the parallelepiped generated by $\mathbf{a},$ $\mathbf{b} + \mathbf{b} \times \mathbf{a},$ and $\mathbf{c}.$

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