
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
We define two sequences of vectors \((\mathbf{u}_n)\) and \((\mathbf{z}_n)\) as follows: First, \(\mathbf{u}_0 = \begin{pmatrix} 2 \\ 2 \end{pmatrix},\) \(\mathbf{z}_0 = \begin{pmatrix} 3 \\ 1 \end{pmatrix}.\) Then for all \(n \ge 1,\) \(\mathbf{u}_n\) is the projection of \(\mathbf{z}_{n - 1}\) onto \(\mathbf{u}_0,\) and \(\mathbf{z}_n\) is the projection of \(\mathbf{u}_n\) onto \(\mathbf{z}_0.\) Find
\[
\mathbf{u}_1 + \mathbf{u}_2 + \mathbf{u}_3 + \dotsb.
\]

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