An unusual inequality condition

Algebra Level 4

Consider all triples of non-zero real numbers that satisfy

\[ \left| \frac { 1 }{ a } \right| +\left| \frac { 1 }{ b } \right| +\left| \frac { 1 }{ c } \right| \le 3 \]

What is the minimum value of

\[ {(a^2+4(b^2+c^2))(b^2+4(a^2+c^2))(c^2+4(a^2+b^2)) ?}\]

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