Factorize the following expression : \[ \left(\frac{x}{2}+y+\frac{z}{3}\right)^3 + \left(\frac{x}{3}-\frac{2y}{3}+z\right)^3 + \left(-\frac{5x}{6}-\frac{y}{3}-\frac{4z}{3}\right)^3 \]
\[
\text{Let }
a=\left(\frac{x}{2}+y+\frac{z}{3}\right),
\]
\[
b=\left(\frac{x}{3}-\frac{2y}{3}+z\right),
\]
\[
c=\left(-\frac{5x}{6}-\frac{y}{3}-\frac{4z}{3}\right)
\]
\[
a+b+c=0
\]
\[
\therefore a^3+b^3+c^3=3abc
\]
\[
=3\left(\frac{x}{2}+y+\frac{z}{3}\right)
\left(\frac{x}{3}-\frac{2y}{3}+z\right)
\left(-\frac{5x}{6}-\frac{y}{3}-\frac{4z}{3}\right)
\]