\(\color{red}f(x),\) \(\color{blue}g(x),\) and the function \({\color{orange}h(x)}={\color{red}f(x)}+{\color{blue}g(x)}\) have fundamental periods of \(3,\) \(4,\) and \(\text{lcm}(3,4)=12,\) respectively.

If \(p(x)\) and \(q(x)\) have fundamental periods \(2\) and \(\pi,\) respectively, what is the fundamental period of \(r(x)=p(x)+q(x)?\)

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