How to draw a continuous line using two seperate existing paths using tikz

I'm trying to draw a curve using two seperate paths. The Line I want to draw should use path=\pathX and use path=\pathY but I don't want it to draw the entire paths which is why I helped myself clipping both paths. I'd also like to join both path segments with lines on either end which I somehow achieved. Yet I would like all four segments to be drawn as one continuous curve unlike what you see in the graphic I attached.

My MWE is as follows.

\documentclass[tikz]{standalone}

\usepackage[]{tikz}
\usetikzlibrary{%
  calc,%
  intersections,%
}%
\begin{document}
\def\dimx{5}
\def\dimy{5}
  \begin{tikzpicture}[]
    \path [name path=I, save path=\pathI] ($.25*(\dimx,0)$) rectangle ($.75*(\dimx,0) + .85*(0,\dimy)$);
    \coordinate [] (p11) at ($0.1*(0,\dimy) + .1*(\dimx,0)$);
    \coordinate [] (p12) at ($0.65*(0,\dimy) + .9*(\dimx,0)$);
    \coordinate [] (p21) at ($0.2*(0,\dimy) + .1*(\dimx,0)$);
    \coordinate [] (p22) at ($0.95*(0,\dimy) + .9*(\dimx,0)$);
    \draw [name path=p1, save path=\pathC]
      (p11) to [out=25, in=220] node [at end, anchor=west] {$p_1$} (p12);
    \draw [name path=p2, save path=\pathA]
      (p21) to [out=35, in=230] node [at end, anchor=west] {$p_2$} (p22);
    \path [name intersections={of=p1 and I}] coordinate (4) at (intersection-2);
    \path [name intersections={of=p1 and I}] coordinate (1) at (intersection-1);
    \path [name intersections={of=p2 and I}] coordinate (3) at (intersection-2);
    \path [name intersections={of=p2 and I}] coordinate (2) at (intersection-1);
    \begin{scope}[]
      \path [clip, use path=\pathI];
      \path [name path=p1, save path=\pathX]
        (p21) to [out=35, in=230]
        node [at end, anchor=west] {$p_1$}
        coordinate [pos=0.15] (A)
        coordinate [pos=0.7] (B)
        (p22);
      \path [name path=p3, save path=\pathY]
        (p11) to [out=25, in=220]
        node [at end, anchor=west] {$p_3$}
        (p12);
      \draw [thick, blue] edge [use path=\pathY] edge [use path=\pathX];
    \end{scope}
    \draw [thick, blue] (1) -- (2) (3) -- (4);
  \end{tikzpicture}
\end{document}

I might be using a completely wrong approach here but that's as far as google and the pgf manual got me. Any suggestions I'll greatly appreciate. Thanks in advance.