Question 11
Determine the largest number $m$ such that the inequality
$$
(a^2 + 4(b^2 + c^2))(b^2 + 4(c^2 + a^2))(c^2 + 4(a^2 + b^2)) \geq m
$$
holds for all real numbers $a, b$ and $c$ not equal to $0$ and satisfying the condition $\left|\frac{1}{a}\right| + \left|\frac{1}{b}\right| + \left|\frac{1}{c}\right| \le 3$.
$$
(a^2 + 4(b^2 + c^2))(b^2 + 4(c^2 + a^2))(c^2 + 4(a^2 + b^2)) \geq m
$$
holds for all real numbers $a, b$ and $c$ not equal to $0$ and satisfying the condition $\left|\frac{1}{a}\right| + \left|\frac{1}{b}\right| + \left|\frac{1}{c}\right| \le 3$.