A specialized security lock requires a sequence of two codes. The probability that the first code is correct is $P(C_1) = 0.8$. If the first code is correct, the probability the second code is correct is $P(C_2 | C_1) = 0.9$. If the first code is incorrect, the probability the second is correct is $P(C_2 | C_1^c) = 0.3$. If the lock opens (meaning both codes are correct), what is the conditional probability that the first code was correct, given that the second code was correct?