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,andtareWFFs • ifwisaWFF,NwisaWFF • 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 1111011 1001000 0101110 0000111 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 p = 0, 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 ApNp ApNq 0 Sample Output tautology not