Chef loves that his cooks involve themselves in group activities. This month, Chef has involved his cooks in a rather very curious game. It involves all his cooks, standing in a circle. Each cook is given one integer to keep, which may be positive or negative (or zero). The sum of all the integers, given to the cooks, is positive.

Let N be a positive integer and S = {1, 2, 3, ..., N}. For a given positive integer d the function f : S --> S is called d-power permutation if there exists a bijection g : S --> S such that g ( g ( ... g ( x ) ... ) ) = f(x) for each x from S , where g is repeated exactly d times.

You are given some bijection f : S --> S and a positive integer D. You need to find the number of those d <= D such that f is d-power permutation.

You are given a sequence of integers A₀, A₁, ..., A_N-1. Initially A_i=i for all i. You need to perform some strange queries with it. Each query has the form "L R D" where 0 <= L <= R < N and D is an integer. If D=0 then you need to find the sum of sines of the numbers A_L, A_L+1, ..., A_R that is sin A_L + ... + sin A_R.

A boolean function is a function of the form f: B_n -> B, where B = {0, 1} and n is a non-negative integer called the arity of the function. Some Boolean functions are projections: p_n^k(x₁, ..., x_n) = x_k. And given an m-ary function f, and n-ary functions g₁, ..., g_m, we can construct another n-ary function: h(x₁, ..., x_n) = f(g1(x₁, ..., x_n), ..., gm(x₁, ..., x_n)), called their composition.