KöMaL 2009 K198

The inhabitants of a certain planet use not four but five basic arithmetic operations. The operations of addition, multiplication, subtraction and division are the same as ours, but they also have a special operation denoted by the operation sign @. We do not know exactly how this operation works, but we have found out that the following properties are valid for all $x$ and $y$: ($a$) $x \mathrel{@} 0 = x$ , ($b$) $x \mathrel{@} y = y \mathrel{@} x$ , ($c$) $(x+1) \mathrel{@} y = (x \mathrel{@} y) + y + 1$. What is the value of $12 \mathrel{@} 5$ on the planet in question?