The following task is a significantly harder version of task Words from the third stage of 16th Polish OI. It wasn't used in the contest itself, but is an extension for those who solved "Words" and want more. :-)Let be a function acting on strings composed of the digits 0 and 1. The function transforms the string by replacing (independently and concurrently) every digit 0 with 1 and every digit 1 with the string . For example , (i.e. assigns an empty string to the empty string). Note that is an injection, or a one-to-one function. By we denote the function composed with itself times. In particular, is the identity function .
We are interested in the strings of the form for This sequence begins with the following strings:
, , , , , .
We call the string a substring of the string if it occurs in as a contiguous (i.e. one-block) subsequence. A sequence of integers is given. Your task is to check whether a string of the form is a substring of for some , and if it is, you shuold find minimal such .
The first line of the standard input contains a single integer , . The second line of standard input holds non-negative integers (), separated by single spaces.
Your programme should print out lines to the standard output, one for each test unit. Your programm should print to standard output minimal non-negative integer , such that is a substring of , or NIE (no in Polish) if such doesn't exist.
For the input data:
2 1 2
the correct result is:
while for following input data:
2 2 0
the correct answer is
Explanation of the example: The string from the first test case is - it is a substring of . In the second test unit there is a string , which is not a substring of for any .
Task author: Bartosz Walczak.