
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
Given vectors $\mathbf{a'} = \begin{pmatrix} 2 \\ -2 \\ 4 \end{pmatrix}$ and $\mathbf{b'} = \begin{pmatrix} 0 \\ 6 \\ -1 \end{pmatrix},$ both are projected onto the same vector $\mathbf{v'}$, resulting in vector $\mathbf{p'}$ for both projections. Additionally, vectors $\mathbf{a'}$, $\mathbf{b'}$, and $\mathbf{p'}$ are collinear. The vector $\mathbf{c} = \begin{pmatrix} 0 \\ 3 \\ -0.5 \end{pmatrix}$ is parallel to $\mathbf{b'}$ and $\mathbf{p'}$ must be perpendicular to $\mathbf{c}$. Find $\mathbf{p'}$.

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