I mean, the answer seems pretty straightforward: Go find some evidence about the answer!
You only said that you don't know the answer, not that it is somehow unknowable.
You could formalize this for something like maximizing expected utility. I don't know the right way to balance all those variables off hand, and that presumably requires some calculus, but basically you'd want to figure out some expected utilities and probabilities:
How much do you want to live?
How long do you have to answer?
How likely is it that you will find the answer in how much time? In other words, how much time does it take to get your chances to better than 50/50 odds.
How much more would you enjoy your life if you just lived it to the deadline instead of spending the time researching and just guessed?
How much does have a chance to keep living close to 50/50 affect your current ability to enjoy your time not researching the answer to the question?
I'm pretty sure if you set all those values there's a function that can spit out exactly how much time you would want to spend researching the answer.