
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
Let two sequences of vectors $(\mathbf{v}_n)$ and $(\mathbf{w}_n)$ be defined as follows: Set $\mathbf{v}_0 = \begin{pmatrix} 2 \\ 1 \end{pmatrix}$ and $\mathbf{w}_0 = \begin{pmatrix} 1 \\ -1 \end{pmatrix}$. For all $n \geq 1$, let $\mathbf{v}_n$ be the result of the projection of $\mathbf{w}_{n-1}$ onto $\mathbf{v}_0$, followed by scaling it down by $\frac{1}{2}$, and let $\mathbf{w}_n$ be the result of the projection of $\mathbf{v}_n$ onto $\mathbf{w}_0$, followed by scaling it up by 2. Compute the sum $\mathbf{v}_1 + \mathbf{w}_1 + \mathbf{v}_2 + \mathbf{w}_2 + \dotsb.$

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