Is Hamlet actually mad?

In the play Hamlet there is a major debate as to whether or not Hamlet is actually “mad” or simply is acting insane for the purpose of deceiving everyone so he can sneakily kill Claudius. At the beginning of the play it seemed quite obvious to me that Hamlet was simply a very good actor but now being in act iv it seems his acting is turning into actual madness. In the beginning of the play when Hamlet first meets his father’s ghost he acts a bit crazy but then becomes very composed in order to devise a plan. One of his first acts of madness was when he went to Ophelia after she had stopped talking to him and did a bunch of emotional actions with his body and did not say anything. To me this simply seemed like an act to deceive Claudius into believing he was simply upset because Ophelia left out of know where. I don’t believe that he was actually mad because he had very witty conversations with all the characters that implied that he wasn’t actually mad. He also told Horatio that he was going to act crazy in the near future. The language he uses when talking to Polonius implies that he is smarter than everyone in the room and that he knows about all their plans to spy on him. This does not seem to be the personality of someone that has gone mad. He seems “crazy” but composed and ha everything planned so that it will go perfectly his way. People who actually go crazy are unorganized and act off impulse. This is how Hamlet starts to act in Acts 3 and 4. When he talks to Ophelia he starts composed but then gets much angrier with her when she lies straight to his face about where her father is. Also in the beginning he had a clear purposed plan to make a play and watch Claudius’s reaction to find if he is guilty or not; however as the play approaches he is not a composed as you might think. When the play is going on he is talking the whole time and saying rude things to both Ophelia and his mother. When he is supposed to be watching Claudius for a reaction he isn’t even patient enough to let his plan do its work he instead spoils the entire play be yelling what happens right to Claudius and Claudius ends the play. This was not characteristic of Hamlet. This is when I believe that he actually begins to go crazy. At first he was simply playing a part but it seems that he has fallen into the part that he was acting and now is actually mad. His madness even grows when he goes to talk to his mother and accidently kills Polonius when he finds him spying. This is the peak of his madness and after this event his metaphors and puns are so crazy they are hard to decipher and analyze and some of them don’t make sense at all. In conclusion Hamlet did not start as mad but has grown in to the part he was playing.