Example 5 Let $x, y, z$ be non-negative real numbers, and satisfy $\sum_{\mathrm{cyc}} x=32$. Try to find the maximum value of $\sum_{\mathrm{cyc}} x^{3} y$.

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Let $x=\max \{x, y, z\}$, then
$$\begin{aligned}
& 27\left(\sum_{\text {cyc }} x\right)^{4}-256\left(\sum_{\text {cyc }} x^{3} y\right) \\
= & z\left(148\left(x z(x-z)+y^{2}(x-y)\right)+108\left(y z^{2}+x^{3}\right)\right. \\
& \left.+324 x y(x+z)+27 z^{3}+14 x^{2} z+162 y^{2} z+176 x y^{2}\right) \\
& +(x-3 y)^{2}\left(27 x^{2}+14 x y+3 y^{2}\right) \geqslant 0,
\end{aligned}$$

with equality if and only if $x-3 y=z=0$. Therefore,
$$\sum_{\mathrm{cyc}} x^{3} y \leqslant \frac{27}{256}\left(\sum_{\mathrm{cyc}} x\right)^{4}=110592$$

with equality if and only if $x=24, y=8, z=0$.