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 $a + b + c \le 110\,000\,000$.