Unreliable Witness and Bayes' Theorem

You have a pet cat that sometimes sneaks out of the house while you're at work. Every day, you call your neighbor to ask whether the cat left. Your neighbor always gives a definite yes or no -- but sometimes they lie (saying the cat left when it didn't, or saying it stayed when it actually left). The probability that your cat actually leaves on any given day is $P(C) = 0.001$. The probability that your neighbor lies is $P(L) = 0.1$, independent of whether the cat actually left. Your neighbor tells you the cat left. What is the probability that the cat actually left the house?