Thanks everyone!
mhtaipei: Why did you suggest me to buy the full y fare? You can buy a discounted J ticket for the price of a full y fare.

How do you get on Galileo? I thought it is only for travel agency only?

I don't think I am looking for the unltra cheap Y ticket. I need to buy the H class ticket to qualify for AAdvantage miles. It usually costs more.

Can I still pay the low-season fare if I leave on 12/4 and come back on 12/30?