$2.[20$ points, $10$ points per item$]$ Let L$1$ and L$2$ be the following two languages:

L$1=\{$M$,$x$,$y$$:M isaTMandacceptsatleastoneofxory$\}$ and L$2=\{$M$,$x$,$y$$:M isaTMandacceptsexactlyoneofxory$\}$

$($a$)$ Prove that L$1$ is recognizable. You only need to describe a Turing machine that recognizes L$1.$ No justification for its correctness is needed.

$($b$)$ Prove that L$2$ is not recognizable. $($Hint: Mapping reduction from ATM.$)$