Probability tail from moment sequence

Let $X$ be a random variable such that the moment generating function of $X$ exists in a neighborhood of zero and $$E(X^n) = \frac{(-1)^n 2}{5} + \frac{2^{n+1}}{5} + \frac{1}{5},\quad n=1, 2, 3, \dots.$$ Then, $P\left(\left|X - \frac{1}{2}\right| > 1\right)$ equals