2829：How Many Free Dimensions?

In calculating the outer product of vectors, existence of constraints usually makes it that not all the components of the product are independent of the others. For example, the outer product of two two-dimensional vectors A = (a₁, a₂) and B = (b₁, b₂), C = A × B = (c₁₁, c₁₂, c₂₁, c₂₂). If we set c₁₂ = c₂₁, then only three of the four components of C is independent, or C has three free dimensions.

Now we have k n-dimensional vectors A₁, A₂, …, A_k. The product of them is a vector of n^k components Z = (z_α) where α enumerates all possible subscripts i₁i₂…i_k (1 ≤ i_j ≤ n for all j s.t. 1 ≤ j ≤ k). We can then apply a rule of constraint of the form s₁s₂…s_k=t₁t₂…t_k to the subscripts. Here s₁s₂…s_k and t₁t₂…t_k are two strings consisting of the same set of lowercase letters. A letter appearing in one string will also appear in the other one and it can have multiple occurrences in a string. When a rule is applied, the letters in it are replaced by arbitrary integers between 1 and n (inclusive) provided that the same letters are replaced by the same integers and different letters are replaced by different integers. If a resulted string after replacements is p₁p₂…p_k=q₁q₂…q_k, let α be p₁p₂…p_k and β be q₁q₂…q_k, then we set z_α = z_β. Given the number of vectors and their dimensions and a rule of constraint, you are required to compute the number of free dimensions of the product of the vectors.

The input contains several test cases. Each test case consists of two lines followed by a blank one. On the first line there are two integers which are n and k in the order they appear. On the second line is a rule of constraint. Two zeroes on a separate line follow the last test case.

For each test case, output one line containing the number of free dimensions of the product of vectors.

In the second test case in the sample input, the rule iij=jii represents the constraints z₁₁₂ = z₂₁₁, z₁₁₃ = z₃₁₁, z₂₂₁ = z₁₂₂, z₂₂₃ = z₃₂₂, z₃₃₁ = z₁₃₃ and z₃₃₂ = z₂₃₃.

You can safely assume that all calculations can be done with 64-bit integers.