Can I use neural networks to forecast the next 5000 values?

Question

Suppose I trained a neural network (RNN, LSTM etc.) which takes 60 values as input to forecast the next (61st) value. Now, I have a train set of 10,000 observations and test set of 5,000. This is easily done in SARIMAX. I can easily fit the model on 10,000 rows and predict the next 5,000. But how will I do this in case of the neural network.

Should I just use the model I trained (takes in 60 and predicts the 61st) and give it the last 60 values in my training dataset to predict the first one in the test dataset and use a rolling window of one step? This feels wrong intuitively.