
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 two sets of real numbers $(a, b, c)$ and $(d, e, f)$ such that:
\[\begin{pmatrix} 3 \\ a \\ c \end{pmatrix} \times \begin{pmatrix} 6 \\ b \\ d \end{pmatrix} = \mathbf{0}\]
and
\[\begin{pmatrix} 4 \\ b \\ f \end{pmatrix} \times \begin{pmatrix} 8 \\ e \\ d \end{pmatrix} = \mathbf{0},\]
find the ordered set $(a, b, c, d, e, f)$.

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