When a certain biased die is rolled, a particular face occurs with probability $\displaystyle\frac{1}{16}$ and its opposite face occurs with probability $\displaystyle\frac{1}{2} + x$. All other four faces occur with probability $p$ each. (Opposite faces sum to $7$ on a die.) If $0 < x < \displaystyle\frac{1}{4}$ and the probability of obtaining a total sum of $7$ when such a die is rolled twice, is $\displaystyle\frac{13}{83}$, then the value of $x$ is: