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A binary random variable $X$ takes the value of $1$ with probability $1/3$. $X$ is input to a cascade of $2$ independent identical binary symmetric channels (BSCs) each with crossover probability $1/2$. The output of BSCs are the random variables $Y_{1}$ and $Y_{2}$ as shown in the figure.

The value of $H( Y_{1} )+H( Y_{2} )$ in bits is ______.