Let $a,$ $b,$ $c$ be positive real numbers.  Find the smallest possible value of
\[6a^3 + 9b^3 + 32c^3 + \frac{1}{4abc}.\]
Question:
Let $a,$ $b,$ $c$ be positive real numbers.  Find the smallest possible value of
\[6a^3 + 9b^3 + 32c^3 + \frac{1}{4abc}.\]
Answer:
Let's think step by step.

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6
Let $a,$ $b,$ $c$ be positive real numbers.  Find the smallest possible value of
\[6a^3 + 9b^3 + 32c^3 + \frac{1}{4abc}.\]
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