AMC 12 2022b Problem 9

The sequence $a_0,a_1,a_2,\cdots$ is a strictly increasing arithmetic sequence of positive integers such that $$2^{a_7}=2^{27} \cdot a_7.$$ What is the minimum possible value of $a_2$?

$\textbf{(A) }\ 8 \textbf{(B) }\ 12 \textbf{(C) }\ 16 \textbf{(D) }\ 17 \textbf{(E) }\ 22$