
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
There exist vectors $\mathbf{u}$ and $\mathbf{v}$ such that 
\[\mathbf{u} + \mathbf{v} = \begin{pmatrix} 3 \\ 1 \\ -4 \end{pmatrix},\]
where $\mathbf{u}$ is parallel to $\begin{pmatrix} 2 \\ -1 \\ 1 \end{pmatrix},$ and $\mathbf{v}$ is orthogonal to $\begin{pmatrix} 2 \\ -1 \\ 1 \end{pmatrix}.$ Find $\mathbf{v}.$

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