Generally, rephrase the problem in abstract terms.
Abstraction gets rid of the irrelevent minutiae of the specific problem and gives us the opportunity to solve the more general problem first. There may be a series of increasingly abstract representations of the problem, like a hierarchy classes in OOP.
My (rather vague) understanding of mathematicians is that they have a menagerie of increasingly abstract math objects (sets, groups, categories), and are able to apply proofs from the more abstract objects to any of the derived objects. This seems to work well for them.
So, if possible, try to solve the problem at the highest abstration layer first. If you don't suceed there then try to solve it at the next layer instead. If you do succeed, then you have a general solution that applies to that abstraction layer and all layers below it (possibly with tweaks, like overridden methods in OOP).
Wrt the specific question ("what is the best way to organize a liberal democracy of >100M citizens?"):-
This problem is already fairly abstract, but first, define "best", "organize", and "democracy".
-We can only know a solution is best if we first find all solutions. Depending on the problem, this might not be worth the effort, an aproximate solution might be adequate.
-In this context I assume that 'organise' means getting the population to do something.
-Quite a few countries have elections, are they all democracies?
Some relevent questions to consider are:
-Is the aim to get the population to all agree on an action?
-Is there a clear benefit to a proposed action, or is it subject to unknowns?
-Is the population rational?
-Do they understand the advantages of action?
-Are there any disadvantages of action? Do they understand those?
-In general any solution is likely to have a number of different aspects each with their own costs and benefits (monetary cost, time cost, health cost, ...)
-Are all members of the population equally affected by the advantages and disadvantages, or are subsets affected unequally? This is likely to be unanswerable, given that individuals are likely to assign different costs to different aspects.
-Can they be coerced? (Yes, liberal democracies coerce too! think taxes on carbon fuel for example).
In a problem that compares apples and oranges and shellfish and pyramids, "The best way to organize..." seems to depend on the whim of whoever happens to be pulling the levers of power; there is no best solution to this.