For real numbers $x$, let  
$$P(x)=1+\cos(x)+i\sin(x)-\cos(2x)-i\sin(2x)+\cos(3x)+i\sin(3x)$$
where $i = \sqrt{-1}$. Find the number of values of $x$ with $0\leq x<2\pi$ for which  
$$P(x)=0.$$

Let's think step by step and output the final answer within \boxed{}.
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