They do not recognize that they are being given a great opportunity before the presentation, but afterward they do.
The answer should be moving tense
I HOPED I HELPED!
Present of began is begin.
Past participle of I think you meant "begin" is begun.
It is true that Gertrude is still convinced that Hamlet is mad. As a matter of fact, she thinks her son is mad right until the end of the play, and then they all die.