AMC 12 2024a Problem 25

A graph is $\textit{symmetric}$ about a line if the graph remains unchanged after reflection in that line. For how many quadruples of integers $(a,b,c,d)$, where $|a|,|b|,|c|,|d|\le5$ and $c$ and $d$ are not both $0$, is the graph of $$y=\frac{ax+b}{cx+d}$$symmetric about the line $y=x$?

$\textbf{(A) }1282 \textbf{(B) }1292 \textbf{(C) }1310 \textbf{(D) }1320 \textbf{(E) }1330$