# Test whether words of less a's than b's or c's but not at the same time is context-free

I want to test whether $$L= \{w\in\{a,b,c\}^* \mid |w|_a<|w|_b \text{ or } |w|_a<|w|_c,\text{ but not at the same time} \}$$ is CFL or not (I assume not), but I am struggling to do so.

The closest I have been to prove that it isn't a CFL is by seeing that the languages $$L_1=\{a^nb^{n+1}c^n\mid n\ge0 \}$$ and $$L_2=\{a^nb^nc^{n+1}\mid n\ge0 \}$$, for example, are contained within it and are obviously not context-free, but that doesn't prove anything.

Any tips?

Let me make it more explicit. Suppose $$L$$ is context-free and $$p>0$$ is a pumping length for it as in the pumping lemma for context-free language. Try pumping up or down the word $$a^pb^{p+1}c^p$$ out of $$L$$.
(You can also pump up or down the word $$a^pb^pc^{p+1}$$ out of $$L$$.)