The word in italics "thoroughly" is an adverb. (Option 3).
There are two things that help you identify the word as an adverb. First of all, the word ends in "-ly". This suffix is usually used to create adverbs. However, there are some adjectives that also end in "-ly". So, if you pay attention to what the word "thoroughly" modifies, you should be able to distinguish that it modifies the verb "captivated" and not the noun "dance". If a modifier explains or characterizes the action (the verb), it is an adverb.
