AMC 12 2022a Problem 10

How many ways are there to split the integers $1$ through $14$ into $7$ pairs such that in each pair, the greater number is at least $2$ times the lesser number?

$\textbf{(A) } 108 \textbf{(B) } 120 \textbf{(C) } 126 \textbf{(D) } 132 \textbf{(E) } 144$