
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 $\mathbf{a} = \begin{pmatrix} 3 \\ -6 \\ 2 \end{pmatrix}$ and $\mathbf{c} = \begin{pmatrix} -1 \\ 2 \\ 0 \end{pmatrix}$. Find the vector $\mathbf{b}$ such that $\mathbf{b}$ is orthogonal to $\mathbf{a}$ and $\mathbf{b}$ is collinear with a new $\mathbf{c}$ formed by adding a vector $\begin{pmatrix} 1 \\ -2 \\ 1 \end{pmatrix}$ to the original $\mathbf{c}$.

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