There are several ways to write the number ${\displaystyle \frac{1}{2}}$ as a sum of square reciprocals using distinct integers.

For instance, the numbers $\{2,3,4,5,7,12,15,20,28,35\}$ can be used:

$$\begin{array}{rl}{\displaystyle \frac{1}{2}}& ={\displaystyle \frac{1}{2^2}}+{\displaystyle \frac{1}{3^2}}+{\displaystyle \frac{1}{4^2}}+{\displaystyle \frac{1}{5^2}}+\\ & {\displaystyle \frac{1}{7^2}}+{\displaystyle \frac{1}{{12}^2}}+{\displaystyle \frac{1}{{15}^2}}+{\displaystyle \frac{1}{{20}^2}}+\\ & {\displaystyle \frac{1}{{28}^2}}+{\displaystyle \frac{1}{{35}^2}}.\end{array}$$

In fact, only using integers between $2$ and $45$ inclusive, there are exactly three ways to do it, the remaining two being: $\{2,3,4,6,7,9,10,20,28,35,36,45\}$ and $\{2,3,4,6,7,9,12,15,28,30,35,36,45\}$.

How many ways are there to write ${\displaystyle \frac{1}{2}}$ as a sum of reciprocals of squares using distinct integers between $2$ and $80$ inclusive?