AMC 12 2023a Problem 23

How many ordered pairs of positive real numbers $(a,b)$ satisfy the equation $$(1+2a)(2+2b)(2a+b) = 32ab?$$

$\textbf{(A) }0 \textbf{(B) }1 \textbf{(C) }2 \textbf{(D) }3 \textbf{(E) }\text{an infinite number}$