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.