Let $S$ be a set of size $n$. Let $T_k$ be the set of all ordered k-tuples $(A_1, A_2, A_3, A_4…...A_k)$ where $A_i$ is a subset of $S$, such that $A_1 ∪ A_2 ∪ A_3 ∪ A_4…...A_k = S$.

Given $n=2718281828$ and $k=3141592653$, find the sum of $|A_{1}∩ A_{2} ∩ A_{3} ∩ A_{4}……..∩ A_{k}|^2$ for all k-tuples $(A_1, A_2, A_3, A_4…...A_k)$ belonging to $T_k$. The answer can be huge, so compute it modulo $10^9+7$.