KURSCHAK 1972 Problem 3

$ABCD$ is a square side $10$ . There are four points $P_1, P_2, P_3, P_4$ inside the square. Show that we can always construct line segments parallel to the sides of the square of total length $25$ or less, so that each $P_i$ is linked by the segments to both of the sides $AB$ and $CD$ . Show that for some points $P_i$ it is not possible with a total length less than $25$ .