The intersect tool does most of the work for you (when you choose the line and polygon feature classes as input). It chops up your lines into intersecting segments and dissolves them based on the polygon that chopped it. If you're working with a geodatabase feature class, there will be a length field showing you how much of the line intersects which polygon. There will be a field with the line ID and a field with the polygon ID. Then you can use the Summary Statistics Tool to count the number of segments each line was chopped into (to get the number of polygons a line crossed). For your model, you should just have the intersect tool feeding into the summary statistics tool.
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