
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
Find the unit vector $\mathbf{v}$, lying in the $xz$-plane, that makes an angle of $60^\circ$ with $\begin{pmatrix} 2 \\ 2 \\ -1 \end{pmatrix}$, and an angle of $30^\circ$ with $\begin{pmatrix} 0 \\ 1 \\ -1 \end{pmatrix}$.

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