In Menachot 65a-66a (according to the Sefaria translation): The Boethusians say, Shavuot falls on a Sunday [because they hold that "the day after the Shabbat" refers to the Sunday after Shabbat Chol HaMoed Pesach]. 

What I would like to know specifically is why this group linked the 'day after the Shabat' with the Shabat Chol HaMoed Pesach. Upon which arguments did they build this view? 

I know how their view was refuted, but I can't find any background on how they came to the conclusion - that by their view that the usage of the word Shabat refers to the seventh day - the word Shabat had to mean the Shabat Chol HaMoed Pesach. 

Anyone who could help me with that?