Example. $($Shuffle$)$ We define a new operator on two strings:

$xy={?}_{\text{d}\text{e}\text{f}}{x}_1{y}_1{x}_2{y}_2{x}_3{y}_{4}dots{x}_k{y}_k$dots If $x$ and $y$ have different lengths,

whatever is left is appended to the end of their shuffle. We may define

the shuffle of two languages: $L_1{L}_2={?}_{\text{d}\text{e}\text{f}}\{xy|xin{L}_1,yin{L}_2\}.$

Obviously, $xyyx.x=x=x.$

Assume both $L_1$ and $L_2$ are regular languages. What is the fa for

$L_1{L}_2?$