Christmas Streak 81/88: Limits on Geometry #2

There is a square $\rm A_1B_1C_1D_1$ with one of its side lengths $7.$

For every natural number $n<N,$ conduct these:

1. Let $\rm E_n,~F_n$ be the midpoints of $\rm \overline{B_nC_n},~\overline{C_nD_n}$ respectively.

2. Let $\rm G_n,~B_{n+1}$ be the intersections of $\rm \overline{A_nE_n},~\overline{A_nC_n}$ with $\rm \overline{B_nF_n}.$

3. Pick a point $\rm C_{n+1}$ on $\rm \overline{C_nD_n}$ such that $\rm \overline{A_nC_n}$ is perpendicular to $\rm \overline{B_{n+1}C_{n+1}}.$

4. Pick a point $\rm D_{n+1}$ on $\rm \overline{A_nD_n}$ and $\rm A_{n+1}$ on $\rm \overline{A_nC_n}$ such that $\rm \overline{B_{n+1}C_{n+1}}$ is perpendicular to $\rm \overline{C_{n+1}D_{n+1}}$ and parallel to $\rm \overline{D_{n+1}A_{n+1}}.$

Find the area of the colored area as $N$ approaches infinity.

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This problem is a part of <Christmas Streak 2017> series .