2023 AMC 12A Problem 23

How many ordered pairs of positive real numbers $(a,b)$ satisfy the equation

$(1+2a)(2+2b)(2a+b) = 32ab?$

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

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.