Given L is a regular language, prove that Perm(L) is Context-Free

Given a regular language $$L$$ defined over $$Sigma = {0, 1}$$. We define a new language $$Perm(L) = {w mid exists x in L, w in perm(x)},$$ where $$perm(x)$$ is the set of all permutations of the word $$x$$.

We have to show that $$Perm(L)$$ is context-free.

I have looked at the properties that neither CFL nor Regular languages are closed under permutation. Also, the problem comes with the following hint.

Hint: Build a PDA for $$perm(L)$$ which guesses a permutation of the input string and runs the DFA for $$L$$ on it, and uses the stack to check that it indeed guessed a valid permutation of the input string. The fact that the alphabet has only two symbols will be crucial for the latter check.

I am unable to use this hint to proceed further. Any help is appreciated!