# Time series forecasting when one of the series is known

I have a problem where there are two time series $$\{x_t\}_{t \geq 1}$$ and $$\{z_t\}_{t \geq 1}$$. These two time series are correlated for fixed time instant but uncorrelated with each other across time. We assume that time series $$\{z_t\}_{t \geq 1}$$ is completely known to us. Is there a way to incorporate this knowledge into an LSTM model for multistep prediction of $$\{x_t\}_{t \geq 1}$$. My main idea is that if e.g., I want to predict for $$10$$ time steps ahead (predict $$x_{t+1}, \dots, x_{t+10}$$), I can use a look-back window of size 10 on $$x_{t-10}, \dots, x_{t-1}$$ and on $$z_{t+1}, \dots, z_{t+10}$$, so a feature dimension $$D=2$$. My main concern is what would happen if I wanted to use a longer look-back window on $$\{x_t\}_{t \geq 1}$$ while still wanting to predict the next 10 values of $$\{x_t\}_{t \geq 1}$$. In this case the features of my input would require different look-back windows? Is this possible or is there a trick to bypass this?