HK-PC 2009 Problem 2

Find the total number of solutions to the following system of equations: $$\begin{cases} a^2+bc\equiv a\pmod {37}\\ b(a+d)\equiv b\pmod {37}\\ c(a+d)\equiv c\pmod{37}\\ bc+d^2\equiv d\pmod{37}\\ ad-bc\equiv 1\pmod{37}\end{cases}$$