
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
Define three sequences of vectors $(\mathbf{v}_n)$, $(\mathbf{w}_n)$, and $(\mathbf{u}_n)$ as follows: Initialize $\mathbf{v}_0 = \begin{pmatrix} 2 \\ 2 \end{pmatrix}$, $\mathbf{w}_0 = \begin{pmatrix} 3 \\ 1 \end{pmatrix}$, and $\mathbf{u}_0 = \begin{pmatrix} 1 \\ -1 \end{pmatrix}$. For all $n \geq 1$, 
1. $\mathbf{v}_n$ is the projection of $\mathbf{w}_{n-1}$ onto $\mathbf{v}_0$;
2. $\mathbf{w}_n$ is the projection of $\mathbf{v}_n$ onto $\mathbf{w}_0$;
3. $\mathbf{u}_n$ is the projection of $\mathbf{w}_n$ onto $\mathbf{u}_0$.

Find:
\[\sum_{n=1}^\infty (\mathbf{v}_n + \mathbf{w}_n + \mathbf{u}_n).\]

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