
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 \(\mathbf{v}\) and \(\mathbf{w}\) be vectors such that
\[
\operatorname{proj}_{\mathbf{w}} \mathbf{v} = \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix},
\]
and let \(\mathbf{u}\) be another vector for which 
\[
\operatorname{proj}_{\mathbf{w}} \mathbf{u} = \begin{pmatrix} 0 \\ -1 \\ 2 \end{pmatrix}.
\]
Compute \(\operatorname{proj}_{\mathbf{w}} (\mathbf{v} + \mathbf{u}).\)

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