For any two real numbers $x$ and $y,$ define \[x \star y = ax + by + cxy,\]where $a,$ $b,$ and $c$ are constants. It is known that $1 \star 2 = 3,$ $2 \star 3 = 4,$ and there is a non-zero real number $d$ such that $x \star d = x$ for any real number $x.$ What is the value of $d$?
