UVA 1594 Ducci Sequence

Ducci Sequence

Description

A ducci sequence is a sequence of n-tuples of integers. Given an n-tuple of integers (a1, a2, ... ), a C11>n), the next n-tuple in the sequence are formed by taking the absolute differences of neighboring Integers:

( a1, a2,... , aN) --- (| a1- a2|,| a2- a3|,... ,| aN- a1|)

Ducci sequences either reach a tuple of zeros or fall into a periodic loop. For example, the 4-tuple sequence starting and 8,11,2,7 takes 5 steps to reach the zeros tuple:

(8, one, 2, 7)---(3, 9, 5, 1)---(6, 4, 4, 2)---(2, 0, 2, 4)---(2, 2, 2, 2)---(0, 0, 0, 0).

The 5-tuple sequence starting with 4,2,0,2,0 enters a loop after 2 steps:

(4, 2, 0, 2, 0)---(2, 2, 2, 2, 4)---( 0, 0, 0, 2, 2)---(0, 0, 2, 0, 2)---(0, 2, 2, 2, 2)---(2, 0, 0, 0, 2)---

(2, 0, 0, 2, 0)---(2, 0, 2, 2, 2)---(2, 2, 0, 0, 0)---(0, 2, 0, 0, 2)---(2, 2, 0, 2, 2)---(0, 2, 2, 0, 0)---

(2, 0, 2, 0, 0)---(2, 2, 2, 0, 2)---(0, 0, 2, 2, 0)---(0, 2, 0, 2, 0)---(2, 2, 2, 2, 0)---( 0, 0, 0, 2, 2) --- ...

Given an n-tuple of integers, write a program to decide if the sequence are reaching to a zeros tuple or a per Iodic Loop.

Input

Your program was to read the input from standard input. The input consists of T test Cases. The number of test cases T is given on the first line of the input. Each test case starts with a line containing an integer n(3n), which represents the size of a Tuple in the Ducci sequences. In the following line, n integers is given which represents the n-tuple of integers. The range of integers is from 0 to 1,000. Assume that the maximum number of steps of a ducci sequence reaching zeros tuples or making a loop does not exceed A.

Output

Your program is-to-write to standard output. Print exactly one line for each test case. print 'loop' If the Ducci sequence falls into a periodic LOOP, print 'ZERO' If the Ducci sequence Reache s to a zeros tuple.

The following shows sample input and output for four test cases.

Sample Input

4 4 8 11 2 7 5 4 2 0 2 0 7 0 0 0 0 0 0 0 6 1 2 3 1 2 3

Sample Output

Zero Loop Zero loop

UVA 1594 Ducci Sequence