It seems "believe" has a meaning of "think" here. How should I decide which one to use, "believe" and "think"? Is there any difference in nuance?
Yes, "believe" here means "think". "Believe" and "think" have slightly different nuances. "Believe" has a gentle connotation, and is used when the speaker is less sure about the statement they are making. "Think" is used when the speaker is more confident about the statement they are making. Ex: I don't believe I have to work tomorrow, but I'm not sure. I will check again. Ex: I think this jacket is too small, I will try on another one.