The sorcerer is crafting a wand for himself that contains $101$ pearls in a specific sequence. Each pearl is magically generated and can be either $red$ or $blue$ in color, with an equal probability for each. The power of the wand is the maximum number of consecutive pearls of the same colour. The expected power of the wand is $x$.

$x$ can be represented as $\frac{p}{q}$ such that $\text{gcd}(p,q) = 1$. Find $(p \cdot q^{-1})\:mod\:1234567891$.