How does `tput` detect color support

I am trying to make my shell script supporting as much terminals as possible, that is, adding ANSI colors, bold and dim only when supported. However, I want to detect the number of colors supported. We can use tput colors. However, I find that some systems that are stripped down to minimal does not have it installed. So, I want to implement it in shell. How does tput colors work? Any equivalent of tput colors with POSIX commands? Please help and answer.