10. In the cube $A B C D-A_{1} B_{1} C_{1} D_{1}$, points $M$ and $N$ lie on segments $A B$ and $B B_{1}$ (excluding the endpoints of the segments), and $A M=B_{1} N$. Then the range of the angle formed by $A_{1} M$ and $C_{1} N$ is $\qquad$
10. $\left(\frac{\pi}{3}, \frac{\pi}{2}\right)$.

Take points $P, Q$ on $DC, A_{1}A$ respectively, such that
$$
D P=A_{1} Q=A M=B_{1} N \text {. }
$$

Let $A M=x, A B=a(0\frac{a}{a}+\frac{a}{a}>2$
$$
\begin{array}{l}
\Rightarrow 0<\cos \angle P D_{1} Q<\frac{1}{2} \\
\Rightarrow \frac{\pi}{3}<\angle P D_{1} Q<\frac{\pi}{2} .
\end{array}
$$