
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 $\mathbf{a} = \begin{pmatrix} 2 \\ 1 \\ -1 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 1 \\ -2 \\ 3 \end{pmatrix}$. Find the vector $\mathbf{v}$ that satisfies $2\mathbf{v} \times \mathbf{a} = \mathbf{b} \times \mathbf{a}$ and $3\mathbf{v} \times \mathbf{b} = \mathbf{a} \times \mathbf{b}$.

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