Gosha - Nested Binary Search

Apply Aliens Trick twice. Add penalty $\lambda_1$ per Poke Ball and $\lambda_2$ per Ultra Ball. For each Pokemon $i$, compute the best action among: use neither, use Poke Ball only ($+\lambda_1$), use Ultra Ball only ($+\lambda_2$), use both ($+\lambda_1 + \lambda_2$).

Binary search on $\lambda_1$ to use exactly $a$ Poke Balls. Inside that, binary search on $\lambda_2$ to use exactly $b$ Ultra Balls. Total: $O(n \log^2 C)$.