A rescue team allocates 10 hours of search time between Region $A$ (probability of containing the target is $P(A) = 0.6$) and Region $B$ ($P(B) = 0.4$). The detection probability in $A$ after $t_A$ hours is $1 - e^{-0.2 t_A}$, and in $B$ after $t_B$ hours is $1 - e^{-0.1 t_B}$. Which allocation $(t_A, t_B)$ maximizes the overall probability of finding the target under constraint $t_A + t_B = 10$?