6. In a square $ABCD$ with side length 1, points $M$, $N$, $O$, $P$ are on sides $AB$, $BC$, $CD$, $DA$ respectively. If $AM=BM$, $DP=3AP$, then the minimum value of $MN+NO+OP$ is $\qquad$.

Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly.
6. As shown in Figure 2, we have
$$
\begin{array}{l}
M N+N O+O P \\
=M N+N O_{1}+O_{1} P_{2} \\
\geqslant M P_{2},
\end{array}
$$

and the equality holds when points $N, O_{1}$ are both on the line segment $M P_{2}$. Then
$$
\begin{array}{l}
M P_{2}=\sqrt{M A_{1}^{2}+A_{1} P_{2}^{2}} \\
=\sqrt{\left(\frac{3}{2}\right)^{2}+\left(\frac{7}{4}\right)^{2}}=\frac{\sqrt{85}}{4}
\end{array}
$$