3559：Trigonometric optimization

Many problems arising in practical applications may be stated as optimization problems. Usually it is necessary to maximize or minimize so called criterion function taking into account some constraints.

Let’s consider a trigonometric optimization problem. It is necessary to maximize or to minimize criterion function F₁(x) + F₂(y) + F₃(z) with constraint x + y + z = S, where x, y, z – variables, S – parameter, x, y, z, S – natural numbers excluding zero.  Each of the functions F₁, F₂ and F₃ is a trigonometric function sin or cos.

You need to write a program which solves the trigonometric optimization problem.

The input contains 5 lines. The first line describes function F₁ and contains either sin or cos. The second and the third lines describe functions F₂ and F₃ respectively and have the same format as the first line. Next, the fourth line contains either min or max. If the line contains min than it is necessary to minimize criterion function, otherwise it is necessary to maximize criterion function. Finally, the fifth line contains parameter S value (3 ≤ S ≤ 1 000 000).

The only line of the output contains one real number – the found value of the criterion function, described in the input file. Absolute error of your answer must not exceed 10⁻¹⁰.

Beware that GNU compilers for C and C++ provide relatively slow trigonometric functions.