10. (14 points) As shown in Figure 1, in the triangular prism $A B C-A_{1} B_{1} C_{1}$, it is known that the side edge $A A_{1} \perp$ plane $A B C$, and $\triangle A B C$ is an equilateral triangle with a side length of 2. $M$ is a point on $A A_{1}$, $A A_{1}=4, A_{1} M=1, P$ is a point on the edge $B C$, and the shortest distance from $P$ along the side surface of the prism through the edge $C C_{1}$ to point $M$ is $3 \sqrt{2}$. Let this shortest distance intersect $C C_{1}$ at point $N$.
(1) Prove: $A_{1} B / /$ plane $M N P$;
(2) Find the tangent value of the dihedral angle (acute angle) formed by plane $M N P$ and plane $A B C$.
10. (1) From $A A_{1} \perp$ plane $A B C$ and $\triangle A B C$ being an equilateral triangle, we know that all the side faces are congruent rectangles.

As shown in Figure 3, rotate the side face $B C_{1}$ by $120^{\circ}$ so that it lies in the same plane as the side face $A C_{1}$. Point $P$ moves to the position of $P_{1}$, and connect $M P_{1}$. Then $M P_{1}$ is the shortest path from point $P$ along the lateral surface of the prism through $C C_{1}$ to point $M$.
Let $P C=x$. Then $P_{1} C=x$.
In the right triangle $\triangle M A P_{1}$, note that,
$$
\begin{array}{l}
(2+x)^{2}+3^{2}=18 \\
\Rightarrow x=1 .
\end{array}
$$

Thus, $P$ is the midpoint of $B C$.
Therefore, $N C=1$.
Let $A_{1} C$ intersect $M N$ at point $Q$. Then $Q$ is the midpoint of $A_{1} C$.
So, $A_{1} B / / P Q$.
Hence, $A_{1} B / /$ plane $M N P$.
(2) As shown in Figure 3, connect $C H$ and $P P_{1}$. Then $P P_{1}$ is the intersection line of plane $M N P$ and plane $A B C$.
Draw $N H \perp P P_{1}$ at point $H$.
Since $C C_{1} \perp$ plane $A B C$, it follows that $C H \perp P P_{1}$.
Therefore, $\angle N H C$ is the plane angle of the dihedral angle formed by plane $N M P$ and plane $A B C$.
In the right triangle $\triangle P H C$, we have
$$
\angle P C H=\frac{1}{2} \angle P C P_{1}=60^{\circ} \text {, }
$$

Then $C H=\frac{1}{2}$.
In the right triangle $\triangle N C H$,
$$
\tan \angle N H C=\frac{N C}{C H}=2 .
$$