Always anchor high. Humans are highly susceptible to anchoring, so even if you ask for an outlandish sum, and it's above the range, there's a greater chance that they'll come back to you with a salary that they're willing to pay rather than drop you altogether. And the salary they offer will be higher than if you asked for a lower sum.
This comes from the Handbook on the Psychology of Pricing. If you want a raise of $15k, ask for $30k. If you want $113k, ask for $120k. If the top of their range is $80k, they'll tell you, and that's when you get to have fun negotiating.
"Hmm, well, that is lower than what I'm hearing from other companies I'm interviewing with, but tell me more about your benefits. Things like 401k match, vacation time, and how much you pay towards health insurance premiums are worth a lot and I may be willing to accept a lower salary if those benefits are good."
Let's say you're then comparing a $113k job with 10 days off, 2% 401k match, and health insurance premiums of $400/month against a $81k job with 40 days off, a 6% 401k match, and $0/month in health insurance premiums...
Anchor high, then negotiate. Most everything can be negotiated, and they'll tell you if something can't be. If you get disqualified for trying to have the conversation, you probably don't want to work there.