
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 sequences of vectors $(\mathbf{v}_n)$ and $(\mathbf{u}_n)$ such that $\mathbf{v}_0 = \begin{pmatrix} 2 \\ 1 \end{pmatrix}$ and $\mathbf{u}_0 = \begin{pmatrix} 5 \\ 2 \end{pmatrix}$. Then, for all $n \ge 1$, $\mathbf{v}_n$ is the projection of $\mathbf{u}_{n-1}$ onto $\mathbf{v}_0$, and $\mathbf{u}_n$ is the projection of $\mathbf{v}_n$ onto $\mathbf{u}_0$. Find the sum $\mathbf{u}_1 + \mathbf{u}_2 + \mathbf{u}_3$.

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