$$
\begin{array}{l}
M=\left\{x \mid x^{2}-8 x+k<0\right\}, \\
N=\left\{x \mid x^{2}-6 x+5<0\right\}, \\
P=\left\{x \mid x^{2}-10 x+16<0\right\},
\end{array}
$$

satisfy "if $a \in M$, then $a \in N \cup P$". Find the range of the real number $k$.
$$
\begin{array}{l}
M: 4-\sqrt{16-k}k \geqslant 7, \\
16>k \geqslant 0 .
\end{array}\right.\right.
\end{array}
$$

Solving the inequalities corresponding to sets $M$, $N$, and $P$ yields $7 \leqslant k<16$.
Therefore, the range of real number $k$ is $\{x \mid 7 \leqslant k<16\}$.