There are $$$n$$$ points $$$P_0,P_1,\dots,P_{n-1}$$$ on a plane, and the coordinates of $$$P_{i}$$$ are $$$(x_i,y_i)\ (0\le i \lt n)$$$.

Let $$$M=(\sum_{0\le i \lt n} x_i, \sum_{0\le i \lt n} y_i)/n$$$, i.e., the mass center of the initial $$$n$$$ points. The following operations will repeat for $$$k$$$ times:

• For each $$$0\le i \lt n$$$, extend segment $$$MP_i$$$ to $$$Q_i$$$ such that $$$|MP_i|=\cos(2\pi/n)\cdot |MQ_i|$$$.
• Let $$$Q_{-1}=Q_{n-1}$$$ and $$$Q_n=Q_0$$$. For each $$$0\le i \lt n$$$, replace $$$P_i$$$ with $$$R_i$$$, the mass center of $$$\triangle P_iQ_{i-1}Q_{i+1}$$$.

You can see the picture below for better understanding.

Constructing $$$R_i$$$ from $$$P_{i-1}$$$, $$$P_i$$$ and $$$P_{i+1}$$$

Let $$$P'_i=\lim_{k\to\infty} P_i\ (0\le i \lt n)$$$ and $$$P'_n=P'_0$$$. It can be proved that $$$P'_i$$$ will always exist when $$$n\ge 7$$$. Now you need to output the value of

$$$$$$ \sum_{0\le i \lt n} \mathop{MP'_i}^{-\hspace{-2pt}-\hspace{-2pt}-\hspace{-2pt}\rightarrow} \times \mathop{MP'_{i+1}}^{-\hspace{-2pt}-\hspace{-2pt}-\hspace{-2pt}\rightarrow\ \ \ } $$$$$$

P.S. You can try to prove that all $$$P'_i$$$ will be on a certain ellipse though it may have nothing to do with the problem.