In an agricultural experiment regressing crop yield ($Y$ in kg) on fertilizer used ($X$ in kg), the summary statistics are $\bar{x} = 40$, $\bar{y} = 250$, $\sum (x_i - \bar{x})^2 = 800$, and $\sum (x_i - \bar{x})(y_i - \bar{y}) = 3200$. Determine the predicted crop yield when the fertilizer used is $45$ kg.