Use the Pumping Lemma to show $\Sigma^*\setminus\{0^n1^n: n\geq 0\}$ is not regular

Question: Use the Pumping Lemma to show $$L_1 = \Sigma^*\setminus\{0^n1^n: n\geq 0\}$$ is not regular, for $$\Sigma=\{0,1\}$$.

My thoughts: I understand that $$L_2 = \{0^n1^n: n\geq 0\}$$ can be shown to be not regular using the Pumping Lemma by starting with a string in $$L_2$$ of length at least the pumping length, and try to pump it outside the language.

So I've been trying, for $$L_1$$, to start with something and pump it to $$L_2$$, but I have been struggling to find a string such that all it's recompositions can be pumped into $$L_2$$. As it seems whether I can perform the pumping successfully always depends on the decomposition...

If $$L_1$$ was regular, then its complement, $$L_2$$, would be regular. This is a contradiction because you have shown that $$L_2$$ was not regular.