Question 164
The sides of triangle $ABC$ have lengths $a, b, c$ which satisfy
$$
\frac{a^6}{2b^3 + 2c^3} + \frac{b^6 + 2b^3c^3 + c^6}{2a^3} + \frac{2a^3b^3 + 2a^3c^3}{a^3 + b^3 + c^3} = a^3 + b^3 + c^3.
$$
Determine if one of the internal angles of triangle $ABC$ can be a right angle or an obtuse angle, or if all internal angles must be acute.
$$
\frac{a^6}{2b^3 + 2c^3} + \frac{b^6 + 2b^3c^3 + c^6}{2a^3} + \frac{2a^3b^3 + 2a^3c^3}{a^3 + b^3 + c^3} = a^3 + b^3 + c^3.
$$
Determine if one of the internal angles of triangle $ABC$ can be a right angle or an obtuse angle, or if all internal angles must be acute.