Statistical inference problem - complete sufficient and UMVU estimator

螢幕快照 2018-11-11 下午10.15.28.png

螢幕快照 2018-11-11 下午10.15.41.png
螢幕快照 2018-11-11 下午10.15.49.png

I am able to do part (a) by finding the Fisher's Information and using the theorem of Information Inequality.
However, I am very confused about part (b) and (c), as I have no idea what the distribution of S is (Bernoulli can be summed with ZT-Poisson?)
If anyone could provide solution with explanation, that would be great.
This is a question from my assignment that I was not required to complete, hence no solution was provided afterwards, but I would really want to understand it for my own revision. Thanks a lot.