
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
Consider three points whose position vectors are given by $\mathbf{a} = \begin{pmatrix} 1 \\ 0 \\ a \end{pmatrix}, \mathbf{b} = \begin{pmatrix} b \\ 2 \\ 0 \end{pmatrix},$ and $\mathbf{c} = \begin{pmatrix} 9c \\ 9c \\ -2c \end{pmatrix}$, where $a, b,$ and $c$ are real numbers. Determine all possible values of $c$ for which these points lie on a straight line.

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