You MUST file an appeal within 4 months of the decision.This process is called an Article 78 Proceeding. It is done in New York State Supreme Court. For more information you should talk to an attorney.
Remember, NYCHA can have you in Housing Court for a nonpayment case and at the same time be doing a separate termination process at 250 Broadway. It is important that you understand which papers are for which case so that you don’t get your appointments and your defenses for the two separate cases mixed up.