Tautology

WFF ’N PROOF is a logic game played with dice. Each die has six faces representing some subset of the possible symbols K, A, N, C, E, p, q, r, s, t. A Well-formed formula (WFF) is any string of these symbols obeying the following rules:

  • p, q, r, s, and t are WFFs

  • if w is a WFF, Nw is a WFF

  • if w and x are WFFs, Kwx, Awx, Cwx, and Ewx are WFFs.

The meaning of a WFF is defined as follows:

  • p, q, r, s, and t are logical variables that may take on the value 0 (false) or 1 (true).

  • K, A, N, C, E mean and, or, not, implies, and equals as defined in the truth table below.

Definitions of K, A, N, C, and E

w

x

Kwx

Awx

Nw

Cwx

Ewx

1

1

1

1

0

1

1

1

0

0

1

0

0

0

0

1

0

1

1

1

0

0

0

0

0

1

1

1

A tautology is a WFF that has value 1 (true) regardless of the values of its variables. For example, ApNp is a tautology because it is true regardless of the value of p. On the other hand, ApNq is not, because it has the value 0 for $\text {p}=0$, $\text {q}=1$.

You must determine whether or not a WFF is a tautology.

Input

Input consists of several test cases. Each test case is a single line containing a WFF with no more than 100 symbols. A line containing 0 follows the last case.

Output

For each test case, output a line containing “tautology” or “not” as appropriate.

Sample Input 1 Sample Output 1
ApNp
ApNq
0
tautology
not