BALTIC-WAY 2018 Problem 11

The points $A, B, C, D$ lie, in this order, on a circle $\omega$, where $A D$ is a diameter of $\omega$. Furthermore, $A B=B C=a$ and $C D=c$ for some relatively prime integers $a$ and $c$. Show that if the diameter $d$ of $\omega$ is also an integer, then either $d$ or $2 d$ is a perfect square.