An occasionally unreliable professor has devoted his last book to a certain binary operation $*$. When this operation is applied to any two integers, the result is again an integer. The operation is known to satisfy the following axioms: $\text{a})\ x*(x*y)=y$ for all $x,y\in\mathbb{Z}$; $\text{b})\ (x*y)*y=x$ for all $x,y\in\mathbb{Z}$. The professor claims in his book that $1.$ The operation $*$ is commutative: $x*y=y*x$ for all $x,y\in\mathbb{Z}$. $2.$ The operation $*$ is associative: $(x*y)*z=x*(y*z)$ for all $x,y,z\in\mathbb{Z}$. Which of these claims follow from the stated axioms?