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This is an 'adverb infinitive'. It is not a participle, either present of past, because it doesn't end in -ing. And it is not an 'adjective infinitive' since its meaning is adverbial, 'you must capture or trap the other king' IN ORDER 'to win a game of chess'. Adjective has to modify a noun, which isn't the case in this sentence.
The first box is a dangling modifier. The second is a clause. The last is a phrase. (Answers going from top to bottom) :)
What are we reading, you don’t have an image
