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Let $h[n]$ be the impulse response of a discrete-time linear time invariant(LTI) filter. The impulse response is given by $$h[0]=\frac{1}{3}; \, h[1]=\frac{1}{3}; \, h[2]=\frac{1}{3} \text{; and } h[n]=0 \text{ for } n<0 \text{ and } n>2.$$ Let $H(\omega)$ be the discrete-time Fourier transform (DTFT) of $h[n]$, where $\omega$ is the normalized angular frequency in radians. Given that $H(\omega_{0})=0$ and $0< \omega_{0} < \pi$, the value of $\omega_{0}$ (in radians) is equal to__________.
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