input: Write a function in Python that uses a Deep Learning model to predict stock prices based on historical data. The function should load the historical stock data from a CSV file, preprocess the data, split it into training and testing sets, build an LSTM-based model, train the model, and return the predicted stock prices for the test data. The function should be named `predict_stock_prices`.
  
  Example Usage:
  
  from keras.models import Sequential
  from keras.layers import Dense
  from keras.layers import LSTM
  from keras.layers import Dropout
  import numpy as np
  
  # define constants
  time_steps = 60
  batch_size = 32
  
  def predict_stock_prices(filename):
      # build the model
      model = Sequential()
      model.add(LSTM(units=50, return_sequences=True, input_shape=(time_steps, 1)))
      model.add(Dropout(0.2))
      model.add(LSTM(units=50, return_sequences=True))
      model.add(Dropout(0.2))
      model.add(LSTM(units=50))
      model.add(Dropout(0.2))
      model.add(Dense(units=1))
  
      model.compile(optimizer='adam', loss='mean_squared_error')
  
      # read the data
      dataset = np.loadtxt(filename, delimiter=',')
      data = dataset[:, 0:1]
  
      # create train and test data
      x_train = data[:int(len(data)*0.8)]
      x_test = data[int(len(data)*0.8):]
  
      # reshape the data for input format
      x_train = np.reshape(x_train, (x_train.shape[0], time_steps, 1))
      x_test = np.reshape(x_test, (x_test.shape[0], time_steps, 1))
  
      # train the model
      model.fit(x_train, x_train, epochs=50, batch_size=batch_size, validation_data=(x_test, x_test))
  
      # predict stock prices
      predictions = model.predict(x_test)
      return predictions
  
  assert len(predict_stock_prices('data.csv')) > 0
  
output: def predict_stock_prices(filename):
      from keras.models import Sequential
      from keras.layers import Dense, LSTM, Dropout
      import numpy as np
  
      # build the model
      model = Sequential()
      model.add(LSTM(units=50, return_sequences=True, input_shape=(time_steps, 1)))
      model.add(Dropout(0.2))
      model.add(LSTM(units=50, return_sequences=True))
      model.add(Dropout(0.2))
      model.add(LSTM(units=50))
      model.add(Dropout(0.2))
      model.add(Dense(units=1))
  
      model.compile(optimizer='adam', loss='mean_squared_error')
  
      # read the data
      dataset = np.loadtxt(filename, delimiter=',')
      data = dataset[:, 0:1]
  
      # create train and test data
      x_train = data[:int(len(data)*0.8)]
      x_test = data[int(len(data)*0.8):]
  
      # reshape the data for input format
      x_train = np.reshape(x_train, (x_train.shape[0], time_steps, 1))
      x_test = np.reshape(x_test, (x_test.shape[0], time_steps, 1))
  
      # train the model
      model.fit(x_train, x_train, epochs=50, batch_size=batch_size, validation_data=(x_test, x_test))
  
      # predict stock prices
      predictions = model.predict(x_test)
      return predictions