formal languages – If $L$ is regular then $L^{|2|}={w_1w_2 mid w_1,w_2in L, |w_1|=|w_2|}$ is context-free

I have found a problem about proving whether $L^{|2|}={w_1w_2 mid w_1,w_2in L, |w_1|=|w_2|}$ is context-free or not, knowing that $L$ is regular

So far I know that:

  • There are examples where $L$ is regular and $L^{|2|}$ is regular (for example $L={a,b}$)
  • There are examples where $L$ is regular and $L^{|2|}$ is not (for example $L={w mid w=a^N text{ or } w= b^N , Nge0}$)

But I am not sure how to prove that it’s context-free regardless of which regular language I use. I have found similar problems with the same language without imposing restrictions on which words to use, but I am not sure if those apply to this one.