Reverse triangle inequality for three summands

Question

Let $a,b,c\in\mathbb C$. Can we then say that $$|a+b+c|\geq\left||a||-|b|-|c|\right|$$

My guess is that since $|b+c|\leq |b|+|c|$ by the triangle inequality we can maybe pulg that into $$|a+b+c|=|a+(b+c)|\geq||a|-|b+c||\geq ||a|-|b|-|c||$$but I'm not sure if that holds. Any comments?

Answer 1

This is false. Take $a=0, b=1, c=-1$.