Problem: Rectangle Shrinking

Part 1: Notations

$$$x$$$: A brick. $$$x_i$$$ denotes the part of $$$x$$$ belonging to the $$$i$$$-th floor. For example, if $$$x.u=1,\,x.d=2$$$, then $$$x_1.u=x_1.d=1$$$, $$$x_2.u=x_2.d=2$$$, and $$$x_1.l=x_2.l=x.l$$$, $$$x_1.r=x_2.r=x.r$$$.

$$$A_i,\,i=1,2$$$: The set of bricks whose $$$u=d=i$$$.

$$$B$$$: The set of bricks whose $$$u=1,\,d=2$$$.

$$$\texttt{Remove}$$$: Remove a brick $$$x$$$.

$$$\texttt{Shrink}$$$: Shrink a brick $$$x$$$ to another non-empty brick.

$$$\texttt{Keep}$$$: Keep a brick $$$x$$$ unchanged.

Note that in our solution we may operate on a part of brick. For example, for a brick $$$x \in B$$$, we may $$$\texttt{Keep} x_1$$$ while $$$\texttt{Remove} x_2$$$.