A walker starts at 0 and each step moves by an amount drawn uniformly from [-3, -2, -1, 1, 2, 3]. Return the expected number of steps until $|\text{position}| \geq L$.

Let $E_x$ be the expected remaining steps from position x. For every interior position:

$$E_x = 1 + \frac{1}{6}\sum_{s} E_{x+s}, \qquad E_y = 0 \text{ when } |y| \geq L$$

That is a linear system in the interior states, so build it and solve it rather than iterating to convergence.

Input

5

Output

6.725490196078429