Representative monarchy
The basic idea is that one family is the royal family and the people get to vote who the king or queen will be. If there is a corrupt king, the people will have one day to vote him out every week after the kings or queens first two months. The king and queen will have restrictions which will be listed below. It is mixed in with a democracy, republic, and monarchy. The king or queen will have advisors and every state or province will have a few representatives who will give ideas to the leader. The king or queen will have full control of the economic system or more.
Restrictions
The king or queen can't change the government unless a new country is formed, which will be formed by the people. (The people can vote for a new government and country every year, which will be maintained at least a full year to see if it would work out.) No genocide, murder, or any of the basic laws that the United States and other countries follow. There will be no need of a revolution because the people can vote the king or queen out or change the government and country. No social programs allowed at all. Because it technically changes the government. No military force or any other service allowed to keep him in power. They can't cheat the voting ballot (add extra votes for him or her) the result will cause death.