Number Theory

Let $o_i = \underbrace{1 \dots 1}_{i\text{ times}}$ be the number that consists of $i$ ones in its decimal representation.

Bobo has an integer $n$. Find a sequence of possibly negative integers $(x_1, x_2, \dots)$ such that $\sum_{i = 1}^{\infty} o_i \cdot x_i = n$ and $\sum_{i = 1}^{\infty} i \cdot |x_i|$ is minimized.

The input consists of several test cases, and it ends at end-of-file. Each test case has one line containing an integer $n$.

For each test case, output the minimum value of $\sum_{i = 1}^{\infty} i \cdot |x_i|$.

• $1 \leq n < 10^{5000}$
• The sum of the number of decimal digits of $n$ over all inputs does not exceed $50000$.

In the first test case, $x_1 = x_2 = 1$ and $x_3 = x_4 = \dots = 0$. The minimum value is $1 \times 1 + 2 \times 1 = 3$.

In the second test case, $x_1 = 0$, $x_2 = -1$, $x_3 = 1$, and $x_4 = x_5 = \dots = 0$. The minimum value is $2 \times 1 + 3 \times 1 = 5$.