Pumping Lemma with Context Free Languages?

I'd begin by picking a slightly better z = a^(m+2)b^(m+1)c^(m), where m is the pumping length. This string is clearly in the language and its length is greater than or equal to m. So, assuming the language is a CFL, the pumping lemma applies to it.

Now, since you know that |vwx| 0, you also know that vwx must consist of (1) only a's, (2) some a's and some b's, (3) only b's, (4) some b's and some c's, or (5) only c's.

Note: After a bit of back-and-forth in the comments, I see that I'm wrong and William's answer is actually correct. I'll leave this answer here so I can point out where my line of reasoning failed.

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