STATISTICS, Finding sample size, population proportions

I have a really hard time solving this kind of problems. I would really appreciate some help before my exam .

A firm is thinking of selling one more product if the proportion of customers who wants to buy it is at least 20%. To examine if this is the case the firm takes a sample of 100 customers and they will reject the null hypothesis (H0) if the population proportion who says they are going to buy the product is at least 27%.

(Some questions are followed but I already know them)

Now the circumstances are changed.

The person thinks that the probability to NOT reject the null hypothesis should be at least 99% if the population proportion is 17%. He also thinks that the probability to reject the null hypothesis should be at least 99% if the population proportion is 27%

QUESTION: What sample size is required under these circumstances?

If it helps the answer is n>= 365

A really have problems with finding out the sample size when its all about population proportions and using the power and error 1 and 2… Can anyone explain how to solve this problem and also maybe help me to understand how I should think when I get these problems?

Many thanks,