Construct a PDA that recognizes $L = \{w : w \neq a^n b^n : n ≥ 0\}$

Question

I'm trying to find the PDA of the above language. I understand that this is the complement of the language
$L_1=\{w : w=a^nb^n : n\geq0\}$
However, I can't understand the idea behind constructing the PDA. Even if I construct the PDA for $L_1$, and convert the non-accepting states to accepting states and vice-versa, the resulting PDA will still not accept all strings belonging to $L$. Any help is appreciated in this regard.