Define the number of “1” in the binary representation of $n$ as $sigma(n)$ , and the number of “0” successive at the end as $tau(n)$. (For example, notice that $(100)_{10}=(1100100)_2$, then $sigma(100)=3$ and $tau(100)=2$.) Evaluate $prod_{n=1}^infty n^{(-1)^{sigma(n)+tau(n)}}.$