input: Write a function to categorize the sentiment of a given text into positive, negative, and neutral sentiments using the NaiveBayesClassifier from nltk. The function should take a text string as input and return one of 'positive', 'negative', or 'neutral' based on the sentiment analysis of the text. If the number of negative sentiments is greater than the number of positive sentiments, return 'negative'. If the number of positive sentiments is greater than the number of negative sentiments, return 'positive'. If the number of negative sentiments is equal to the number of positive sentiments, return 'neutral'.
  
  assert analyze_sentiment("I love this product! It is amazing.") == 'positive'
  assert analyze_sentiment("I hate this product. It is terrible.") == 'negative'
  assert analyze_sentiment("This product is okay.") == 'neutral'
  assert analyze_sentiment("") == 'neutral'
  assert analyze_sentiment("I love this product, but it broke after a few days.") == 'positive'
output: def analyze_sentiment(text):
      from nltk.classify import NaiveBayesClassifier
      from nltk.corpus import subjectivity
      from nltk.sentiment import SentimentAnalyzer
      from nltk.sentiment.util import extract_unigram_feats
  
      n_instances = 100
      subj_docs = [(sent, 'subj') for sent in subjectivity.sents(categories='subj')[:n_instances]]
      obj_docs = [(sent, 'obj') for sent in subjectivity.sents(categories='obj')[:n_instances]]
  
      train_subj_docs = subj_docs[:80]
      test_subj_docs = subj_docs[80:100]
      train_obj_docs = obj_docs[:80]
      test_obj_docs = obj_docs[80:100]
  
      training_docs = train_subj_docs + train_obj_docs
      testing_docs = test_subj_docs + test_obj_docs
  
      sentim_analyzer = SentimentAnalyzer()
      all_words_neg = sentim_analyzer.all_words([mark_negation(doc) for doc in training_docs])
  
      unigram_feats = sentim_analyzer.unigram_word_feats(all_words_neg, min_freq=4)
      sentim_analyzer.add_feat_extractor(extract_unigram_feats, unigrams=unigram_feats)
  
      training_set = sentim_analyzer.apply_features(training_docs)
      test_set = sentim_analyzer.apply_features(testing_docs)
  
      trainer = NaiveBayesClassifier.train
      classifier = sentim_analyzer.train(trainer, training_set)
  
      neg_count = 0
      pos_count = 0
  
      for word in text.split():
          if classifier.classify(extract_unigram_feats(word.split(), unigram_feats)) == 'subj':
              pos_count += 1
          else:
              neg_count += 1
  
      if pos_count > neg_count:
          return 'positive'
      elif neg_count > pos_count:
          return 'negative'
      else:
          return 'neutral'