
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 $(\mathbf{z}_n)$ with initial vectors $\mathbf{u}_0 = \begin{pmatrix} 2 \\ 4 \end{pmatrix}$ and $\mathbf{z}_0 = \begin{pmatrix} 3 \\ 1 \end{pmatrix}$. For $n \geq 1$, $\mathbf{u}_n$ 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 the sum:
\[\mathbf{u}_1 + \mathbf{u}_2 + \mathbf{u}_3 + \dotsb.\]

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