Construct a PDA for the complement of $a^nb^nc^n$ I am wondering if this is even possible, since $\{a^n b^n c^n  \mid n \geq 0\} \not\in \mathrm{CFL}$. Therefore a PDA that can distinguish a word $w\in\{a^n b^n c^n  \mid n \geq 0\}$ from the rest of $\{a^*b^*c^*\}$ might as well accept it, which sounds contradictory to me.

I guess I need to take advantage of the non-deterministic nature of PDAs but I'm out of ideas. If you could offer some advice I would very much appreciate it.
No, this is context-free. To accept $a^nb^nc^n$, you need to make sure that three numbers are equal. To accept $a^*b^*c^* \setminus a^nb^nc^n$, you just need to make sure that you're in one of the following three cases:


The number of $a$s is different from the number of $b$s; or
The number of $a$s is different from the number of $c$s; or
The number of $b$s is different from the number of $c$s.


Write a PDA for each of these cases, then combine them by jumping nondeterministically to each one from the start state.