Most Doctors will not tell you how to prevent anything as it would destroy their future income. Most have no clue anyway.
My understanding is that ovarian cysts occur because the endocrine system is out of balance.
The Iodine supplement will help but probably won't solve the whole problem. Lugol's or Iodoral. I use Iodoral because it is convenient but Lugols is cheaper if you buy from someone on CZ that sells it.
Plants have a lot of hormones in them the same as meat. You should research which ones you are eating and what hormones they provide. For instance: Yams have a lot of progesterone. Kale and beets also have affects on hormones. Everything in moderation.
I don't know the answer but clearly something is out of wack.
Your multi vitamin has biotin in it. Deeper research would reveal to you that Biotin causes hair loss in large doses. Everything must be balanced because they all play off each other. You should only take extra things when it is clearly needed. If you are taking calcium and magnesium you should be taking potassium, boron, and Vit K. They all work together with Vitamin C.
All just my opinion, of-course. Do your own research keeping in mind that people want to sell you something and may not reveal the full truth to you.