Give me the detailed solution for this question.

Fix an alphabet $.$ For any string $w$ with $|w|2,$ let skip$(w)$ be the string obtained by removing the first two symbols of $w.$ Define two operators on languages:

$f_1(L)=\{win{}^{*}$:skip$(w)inL\},$ and

$f_2(L)=\{skip(w)in{}^{*}$:winL$\}$

$($a$)$ Consider $L^{\text{'}}=L(bb{a}^{**})$ over the alphabet $=\{a,b\}.$ Write a regular expression representing $f_1(L^{\text{'}}).$ Write another regular expression representing $f_2(L^{\text{'}}).$

$($b$)$ Claim: for every regular language $L$ the language $f_1(L)$ is regular. Clearly state whether the claim is TRUE or FALSE, and then prove your answer.

$($c$)$ Claim: for every regular language $L$ the language $f_2(L)$ is regular. Clearly state whether the claim is TRUE or FALSE, and then prove your answer.