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Playing Deity, your biggest problem is citizen unhappiness. When using Monarchy, that is usually allayed through martial law. But WLTKD does not trigger city growth, just extra arrows. Rep and Dem allows WLT(Consul/President)D growth, but unhappiness must be dealt with via wonders (Hanging Gardens, Oracle, Shakes, Michelangelo, JSBach), city improvements (Temples, Colosseums, Cathedrals, Marketplaces & Banks) and extra Luxuries. The latter only work if you have some trade arrows generated by terrain and trade routes, so small cities cannot use it at first. Also, a city must be size 3 or greater for it to kick in, which is why you will also see players found a city with a settler or engineer and immediately "join" two others while Rush Buying a Temple.
So at Deity level, use Monarchy to build settlers and improve the land enough to get more trade out of 3-4 tiles per city. When you have gotten the right techs and a decent number of cities, switch to Rep and crank the Luxuries for WLTCD growth. Keep the unhappiness down with Wonders and Improvements till most have maxed out their available growth, then lower the Luxuries to a rate that balances unhappies with happies.
All of this must be done at a time that your empire is not at war. The second unit away from home adds a red face in Rep, and the first one adds two red in Dem. If you have Shakes it takes care of it for one city, but most often you need to bring the army/navy home or change their supporting cities away from those you want to grow. Or bribe some AI or Barb units to become NONEs.
