6. As shown in the figure, in the cube $A B C D-A_{1} B_{1} C_{1} D_{1}$, $M, N$ are the midpoints of edges $A_{1} B_{1}, B B_{1}$ respectively. A section is made through points $D, M, N$ of the cube, and it is known that the section is a polygon $\Gamma$. Then, the cosine value of the interior angle of $\Gamma$ at vertex $D$ is $\qquad$
As shown in the figure, let the section $\Gamma$ intersect the edges $B C, A_{1} D_{1}$ at points $E, F$. Let the edge length of the cube be 3, $C E=x, A_{1} F=y$. Then, by $\triangle D C E \sim \triangle M A_{1} F, \triangle D D_{1} F \sim \triangle N B E$, we get $\frac{x}{y}=\frac{D C}{M A_{1}}=2, \frac{3-y}{3-x}=\frac{D D_{1}}{N B}=2 \Rightarrow x=2, y=1$.
Therefore, $D E=D F=\sqrt{13}, E F=3 \sqrt{2} \Rightarrow \cos \angle E D F=\frac{26-18}{26}=\frac{4}{13}$.