
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 $O$ be the origin. There exists a scalar $k$ such that for any points $A$, $B$, $C$, and $D$, the vector equation
\[4 \overrightarrow{OA} - 3 \overrightarrow{OB} + 6 \overrightarrow{OC} + k \overrightarrow{OD} = \mathbf{0},\]
implies that $A$, $B$, $C$, and $D$ are coplanar. Find $k$.

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