Minimum length regular expression for $\{ w| w\in\{0,1\}^* $ and for every x that is a prefix of w, $|\#1(x)-\#0(x)| \leq2\}$

Question

$L(M) = $ $\{ w| w\in\{0,1\}^* $ and for every x that is a prefix of w, $|\#1(x)-\#0(x)| \leq2\}$

I tried to use some online convertors and only found this regular expression: ((a+(ϵ+a(ab)*b+b(ba)*a)(a(ab)b+b(ba)a)a)(ab)+b(ba))a+a+a(ab)(ϵ+ab)+(ϵ+a(ab)*b+b(ba)a)(a(ab)b+b(ba)a)(a+a(ab)(ϵ+ab+b)+b+b(ba)(ϵ+ba+a)+ϵ)+((b+(ϵ+a(ab)*b+b(ba)*a)(a(ab)b+b(ba)a)b)(ba)+a(ab))b+b+b(ba)(ϵ+ba)+ϵ

Can someone help me find a shorter one.

Thanks!

Answer 1

Hint. When an exercise looks too complicated, try first a simpler version. In this particular instance, first consider the language $$ L_1 = \bigl\{ w \in \{0,1\}^* \mid \text{and for every prefix $p$ of $w$, $\big||p|_1 - |p|_0\big| \leqslant 1$}\bigr\} $$