def RNN_back_prop(X, Y, parameters, cache):
    # Initialize gradients as an empty dictionary
    grads = {}

    # Retrieve from cache and parameters
    (y_hat, a, x) = cache
    W_aa = parameters['W_aa']
    W_ax = parameters['W_ax']
    W_ya = parameters['W_ya']
    b_y = parameters['b_y']
    b = parameters['b']

    # Initialize gradients
    grads['dW_ax'], grads['dW_aa'], grads['dW_ya'] = np.zeros_like(W_ax), np.zeros_like(W_aa), np.zeros_like(W_ya)
    grads['db'], grads['db_y'] = np.zeros_like(b), np.zeros_like(b_y)
    grads['da_next'] = np.zeros_like(a[0])

    # Backpropagate through timesteps
    for t in reversed(range(len(X))):
        dy = np.copy(y_hat[t])
        dy[Y[t]] -= 1
        grads = RNN_back_prop_step(dy, grads, parameters, x[t], a[t], a[t-1])

    return grads, a