A solar panel array generates power $P$ (in kilowatts) based on the sunlight intensity $I$ (in $kW/m^2$) according to $P(I) = k \cdot I^2$. If the system produces $4$ kW at an intensity of $0.5$ $kW/m^2$, what power will it produce when the intensity increases to $0.8$ $kW/m^2$?