A triplet of positive integers $(a,b,c)$ is called a Cardano Triplet if it satisfies the condition:

$$\sqrt[3]{a+b\sqrt{c}}+\sqrt[3]{a-b\sqrt{c}}=1$$

For example, $(2,1,5)$ is a Cardano Triplet.

There exist $149$ Cardano Triplets for which $a+b+c\le 1000$.

Find how many Cardano Triplets exist such that $a+b+c\le 110000000$.