Yes, a government can exist by combining the best of these two methods and making one big system, but the question would be whether it works right or not, and helps a country out. There are a lot of different kinds of government, and any new ones can be made.
I think probably one of the best systems that we could create would be a hybrid of both capitalist and socialist values. I think it could lead to a nicely layered, and fair, set of social classes. I think it would allow people ample opportunity to move from one social class to the next and be afforded equal opportunities at life.
The United States already does this to an extent. In general, free market principles rule American business. However, when big banks were bailed out, the American people owned stakes in several companies. Taking in tax revenue and redistributing it to all Americans is technically socialism, especially when the poor, disabled and elderly all receive benefits in the way of money or insurance.
No, a government cannot exist applying the best combined aspects of capitalism and socialism, because socialism will always creep in a little bit more until it takes over completely. Socialism, no matter how slight, is not the proper role of the government. A government only exists to provide for a common defense against foreign territories, and to act as a police force. Redistribution is not workable, or lasting.
A government can not exist applying the best combined aspects of capitalism and socialism. These are two sides of the spectrum and they can not have it both ways. I think it will always lean one way or the other and combining of the two different approaches will not work.