I can't tell you how often I've been asked "how much should I offer". First, I'll give a few comments/opinions regarding some of your information: 1) I do not know if the assessed value is useful information in your area. My experience is that often those figures are fairly useless. 2) If the comps sold between 79 and125K maybe 77K is low based upon the condition of the home that you are considering.3) How long you intend to stay in the home probably matters. If you plan to stay indefinitely the offer might be different than if you'll have to sell in a few years. It may be very difficult to resell this home if it's in a small rural area. Therefore, you might want to offer really low or not at all. If, on the other hand you're staying forever and you want the home because it's exactly what you need, then there is a higher answer.
The rule of thumb, in my opinion, is " how bad do you want it?" and " will you be mad/upset if you don't get it?"
If you feel that there are others that would suit you just as well offer low. Start at your $77000 and see what happens. Maybe they'll counter and you'll end up somewhere near 90% of asking price. If you must have this home and they have not lowered their price in 88 days, maybe you'll have to make the best offer you can so that someone else doesn't get it. When I, personally, have really wanted a home, I have offered full price or full price with seller paying 3-4K of my closing costs.
I would love to know if the homes that sold above 99K in that area were nicer than the one you are asking about. Another consideration is that if you are financing it, it will have to appraise for what you are paying for it.
You can have an accepted offer at one amount and then renegotiate it after the appraisal, if the appraisal comes in low. You can also renegotiate after the inspections. You may not end up with the home but you may also avoid overpaying