Let the continuous time real periodic signal $x(*)$ be $x(t)=(2co{s}^2({}_{0}t)-1{)}^2.$

$($a$)$ Determine $\{a_k{\}}_{\text{k}\text{i}\text{n}\text{Z}}$ values satisfying

$x(t)={\displaystyle \underset{k=-}{\overset{}{}}}{a}_k{e}^{jk{}_{0}t}.$

$($b$)$ What is the fundamental period of $x(*)?$

$($c$)$ Determine the output of the LTI system with the impulse response $h_2(t)=e^{-2t}u(t)$ when the input is $x(t).$