Let us define a geometric triangle as an integer sided triangle with sides $a\le b\le c$ so that its sides form a geometric progression, i.e. $b^2=a\cdot c$

An example of such a geometric triangle is the triangle with sides $a=144$, $b=156$ and $c=169$.

There are $861805$ geometric triangles with perimeter $\le {10}^6$.

How many geometric triangles exist with perimeter $\le 2.5\cdot {10}^{13}$?