Statistical inference problem - complete sufficient and UMVU estimator

#1
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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.