Die Throw Sequences - Proposed New Problem

Thank you, Clive! the problem is ready! I hope to try it myself closer to the evening :)

UPD very, very strange feeling, as often. I came from school about 17:30 local time and send successful submission about 20:20 probably. Of course not the whole time was dedicated to the problem - I drank tea twice and fed cats, but mainly I was banging my head against the puzzle. Here is the picture of the page with my attempts of manually analysing the task. Probably it is so confused that hardly can spoil anything. There are attempts to analyze sequence of coin tosses, there are transitional diagrams and equations I derived from them. I thought I can automate equations building and solving thought it looked very laborious approach. Somehow then I managed to catch certain rule, according to which I composed my solution and surprisingly it worked. I still don't understand it perfectly though.

In my head the task found relation to KMP, and to our previous Duel Chances task. However both relations are somewhat weak and not extremely helpful.

Hopefully later you or someone else would kindly describe the idea as I suspect there is some clever and simple explanation which I'm still missing. Meanwhile even despite I don't well understand the solution I'm very proud I did this even though it took over two hours. Thanks a lot it was enjoying in the end :)

BTW congratulations to Artyom who managed to make it first!