
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 $(\mathbf{v}_n)$ and $(\mathbf{w}_n)$ with initial vectors $\mathbf{v}_0 = \begin{pmatrix} 2 \\ 1 \end{pmatrix},$ $\mathbf{w}_0 = \begin{pmatrix} 0 \\ 5 \end{pmatrix}.$ Then, for all $n \geq 1,$ $\mathbf{v}_n$ is the projection of $\mathbf{w}_{n-1}$ onto $\mathbf{v}_0,$ and $\mathbf{w}_n$ is the projection of $\mathbf{v}_n$ onto $\mathbf{w}_0.$ Find:
\[\mathbf{v}_1 + \mathbf{v}_2 + \mathbf{v}_3 + \mathbf{v}_4.\]

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