
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 $\mathbf{a} = \begin{pmatrix} 5 \\ -3 \\ -6 \end{pmatrix}$ and $\mathbf{c} = \begin{pmatrix} -3 \\ 0 \\ 3 \end{pmatrix}.$ Find the vector $\mathbf{b}$ such that $\mathbf{a}, \mathbf{b},$ and $\mathbf{c}$ are collinear, and $\mathbf{b}$ bisects the angle between $\mathbf{a}$ and $\mathbf{c}.$

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