Using American Floors and Caps

My understanding of the european cap/floor is that the exercise of one caplet/floorets does not preclude the exercise of another caplet/flooret in a chain of caplets/floorets. But for American caplets/floorets, once I have exercised the first caplet based on a positive valuation, even though its a 4-period cap, wouldn’t the contract end there?

Building up on the same question: For a company looking to hedge its exposure to rates on variable loan, it’s easy to match the interest payments on the its variable loans with payments from cap/floor. How would a Company use an American Cap/Floor?