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beware spoilers ahead

Firstly, if you outright refuse their request, they'll leave on a more hopeful note; 

And, if acoite's insight is 35 or higher OR his affinity to "the abyss" 10 or higher (which can only be unlocked if acoite was a bad kitty on day 6 and (murdered the pizza guy) is currently on godly probation) then you can actually talk to lich a little further without making them cry;