
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 vectors \(\mathbf{a} = \begin{pmatrix} 2 \\ -2 \\ 4 \end{pmatrix}\) and \(\mathbf{b} = \begin{pmatrix} 1 \\ 6 \\ -1 \end{pmatrix}\) be projected onto the same vector \(\mathbf{v}\), resulting in vector \(\mathbf{p}\) in each case. Additionally, assume vectors \(\mathbf{a}, \mathbf{b},\) and \(\mathbf{p}\) are collinear. Determine vector \(\mathbf{p}\).

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