
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 $\mathbf{b} = \begin{pmatrix} 3 \\ 0 \\ 3 \end{pmatrix}$, both are projected onto the same vector $\mathbf{v}$ resulting in vector $\mathbf{p}$. Furthermore, vectors $\mathbf{a}, \mathbf{b},$ and $\mathbf{p}$ are collinear. Find $\mathbf{p}$.

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