Finding k for Double Root in Polynomial

Consider the polynomial function $$f(x) = 2x^4 - 8x^3 + 6x^2 - 12x + 1.$$

To find the value of $$k$$ for which the polynomial $$g(x) = f(x) + k$$ has a double root, you need to satisfy the condition that both $$g(r) = 0$$ and $$g'(r) = 0$$ for some root $$r$$. Determine the value of $$k$$ such that $$g(x)$$ has exactly one double root. Express your answer in the grid-in format as a reduced fraction or decimal.