Mila is baking a traditional pastry, baklava! Now that she has layered many strips of thin dough, butter, and nuts, she wants to cut the baklava in an ornate way (even if it isn't optimal for sharing the dish fairly.

Namely, her baklava begins as a square of side length $$$l$$$, and Mila is going to perform $$$k$$$ rounds of cutting. In the first round, she will make four cuts that inscribe a smaller square into the original baklava square; the cuts will be from the midpoints of each side to the midpoints of the sides which are not directly across from them. In each subsequent round, she will perform the same four cuts except with the midpoints of the previous round's smaller inscribed square. A visual depiction of this process is shown below for $$$k = 2$$$, with the first round's cuts depicted in red and the second round's cuts depicted in blue:

Mila wants to see just how unfair this cutting scheme is for sharing the baklava, so she wants to calculate the size of the final square which is inscribed in the baklava after the final round of cuts. Can you help her calculate this area?