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It is an adjective because you can replace it another adjective, (stay blue, stay cold) and it still makes sense. It is not a verb because you can't tell someone to stay jump, or stay run. Also it can't be an adverb in the same way it can't be a verb. No one can stay hastily or stay aggresively. Put is describing the current condition of an object.
An adverb is a part of speech. It is any word that modifies any other part of language: verbs, adjectives (including numbers), clauses, sentences and other adverbs, except for nouns; modifiers of nouns are primarily determiners and adjectives.
It is an adverb because it describes the way you should stay AND it is a verb.Also, an adverb does'nt always have to end with LY.
its an adjective , u can check it from google translate