
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 \(\mathbf{u}\) and \(\mathbf{w}\) be vectors such that \(\operatorname{proj}_{\mathbf{w}} \mathbf{u} = \begin{pmatrix} 3 \\ -1 \\ 4 \end{pmatrix}\). Calculate \(\operatorname{proj}_{\mathbf{w}} (3\mathbf{u} + \mathbf{w}).\)

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