$[24$ points, $12$ points per item$]$ Let $L_1$ and $L_2$ be the following two languages:

is a TM and accepts at least one of $x$ or $y$ and

is a TM and accepts exactly one of $x$ or $y$

$($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.