Tessellate S.T.E.M.S. (2019) - Mathematics - Category A (School) - Set 3 - Objective Problem 3

A $2^{2019} + 1$ by $2^{2019} + 1$ grid has some black squares filled. The filled black squares form one or more snakes on the plane, each of whose heads splits at some points but never comes back together. In other words, for every positive integer $\ n > 2$ , there do not exist pairwise distinct black squares $s_1$ , $s_2$ , \dots, $s_n$ such that $s_i$ and $s_{i+1}$ share an edge for $i=1,2, ..., n$ (here $s_{n+1}=s_1$ ).

What is the maximum possible number of filled black squares?

This problem is a part of Tessellate S.T.E.M.S. (2019)