Checking the sites that look at the major travel booking services and the lowest I have seen this evening is around $4,900:
A bit less if you do not mind having a few stops along the way:
I hope that the pricing I was able to find is something that will work for you in order to fly the months you listed. If you could take a moment to rate my assistance so that I know my research was useful and allow me to assist other customers.
So far this is the lowest fairs offered by the major suppliers of low cost air fairs. What I might suggest is to sign up for sites such as the ones listed above as well as those from TripAdvisor as they can then send you alerts when the pricing drops. You can then go ahead and look at the pricing before anybody else does and book the lower price.
Travel agencies would be a good choice as they often have deals that none of us can get because of their contacts with carriers. Some to inquire with include: