TSTST 2024 Problem 1

For every ordered pair of integers $(i,j)$ , not necessarily positive, we wish to select a point $P_{i,j}$ in the Cartesian plane whose coordinates lie inside the unit square defined by

$$i< x< i + 1, j< y< j + 1.$$

Find all real numbers $c > 0$ for which it's possible to choose these points such that for all integers $i$ and $j$ , the (possibly concave or degenerate) quadrilateral $P_{i,j}P_{i + 1,j}P_{i + 1,j + 1}P_{i,j + 1}$ has perimeter strictly less than $c$ .