The action verb is "went" because its an action. you're going somewhere.
NOUN is the infinitive phrase used in this sentence.
What is an infinitive phrase?
"An infinitive phrase is a group of words consisting of an infinitive, a modifier or the use of pronouns, direct objects, indirect objects or complements of action or state expressed in the infinitive."
The infinitive phrase in this sentence is "to find shelter". The infinite phrase here is used as an object due to the fact that it is governed by the verb "ran". In literature, an object is defined as a noun/pronoun that is governed by a verb or even a preposition.
Hence the correct choice is NOUN.
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