Explain existentialism and quote text from "hope, despair, and memory"

Hamlet is often called a tragic hero who is torn between thought and action. why does he not kill claudius when he sees him in act iii, scene iii? why does he kill polonius in act iii, scene iv? how do these two actions affect your understanding of hamlet as a man of thought or a man of action? your answer should be at least 250 words.