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Problem C
Digits
A googol written out in decimal has $101$ digits. A googolplex has one plus a googol digits. That’s a lot of digits!
Given any number $x_{0}$, define a sequence using the following recurrence:
\[ x_{i+1} = \text {the number of digits in the decimal representation of }x_{i} \]Your task is to determine the smallest positive $i$ such that $x_{i} = x_{i-1}$.
Input
Input consists of several lines. Each line contains a value of $x_{0}$. Every value of $x_{0}$ is non-negative and has no more than one million digits. The last line of input contains the word END. There are at most $10\, 000$ values of $x_0$ and the size of the input file is at most $2$ MB.
Output
For each value of $x_0$, output the corresponding value of $i$.
| Sample Input 1 | Sample Output 1 |
|---|---|
42 5 END |
3 2 |
