The equations are equivalent to \begin{align*} 4x^2 - 4y + 1 &= 0, \\ 4y^2 - 4x + 1 &= 0.\end{align*} Summing these equations gives $$4x^2 - 4y + 1 + 4y^2 - 4x + 1 =0,$$ or $$(4x^2 - 4x + 1) + (4y^2 - 4y + 1) = 0.$$ Factoring the squares of binomials gives $$(2x - 1)^2 + (2y-1)^2 = 0.$$ Since squares are always non-negative, it follows that $$2x - 1 = 2y-1 = 0,$$ so $ x = y = \frac 12 $. The desired answer is $ \frac{1}{\frac 18 + \frac 18} = \boxed{4} $.