CF1254C Point Ordering

This is an interactive problem. Khanh has $ n $ points on the Cartesian plane, denoted by $ a_1, a_2, \ldots, a_n $ . All points' coordinates are integers between $ -10^9 $ and $ 10^9 $ , inclusive. No three points are collinear. He says that these points are vertices of a convex polygon; in other words, there exists a permutation $ p_1, p_2, \ldots, p_n $ of integers from $ 1 $ to $ n $ such that the polygon $ a_{p_1} a_{p_2} \ldots a_{p_n} $ is convex and vertices are listed in counter-clockwise order. Khanh gives you the number $ n $ , but hides the coordinates of his points. Your task is to guess the above permutation by asking multiple queries. In each query, you give Khanh $ 4 $ integers $ t $ , $ i $ , $ j $ , $ k $ ; where either $ t = 1 $ or $ t = 2 $ ; and $ i $ , $ j $ , $ k $ are three distinct indices from $ 1 $ to $ n $ , inclusive. In response, Khanh tells you: - if $ t = 1 $ , the area of the triangle $ a_ia_ja_k $ multiplied by $ 2 $ . - if $ t = 2 $ , the sign of the cross product of two vectors $ \overrightarrow{a_ia_j} $ and $ \overrightarrow{a_ia_k} $ . Recall that the cross product of vector $ \overrightarrow{a} = (x_a, y_a) $ and vector $ \overrightarrow{b} = (x_b, y_b) $ is the integer $ x_a \cdot y_b - x_b \cdot y_a $ . The sign of a number is $ 1 $ it it is positive, and $ -1 $ otherwise. It can be proven that the cross product obtained in the above queries can not be $ 0 $ . You can ask at most $ 3 \cdot n $ queries. Please note that Khanh fixes the coordinates of his points and does not change it while answering your queries. You do not need to guess the coordinates. In your permutation $ a_{p_1}a_{p_2}\ldots a_{p_n} $ , $ p_1 $ should be equal to $ 1 $ and the indices of vertices should be listed in counter-clockwise order.

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The image below shows the hidden polygon in the example: The interaction in the example goes as below: - Contestant reads $ n = 6 $ . - Contestant asks a query with $ t = 1 $ , $ i = 1 $ , $ j = 4 $ , $ k = 6 $ . - Jury answers $ 15 $ . The area of the triangle $ A_1A_4A_6 $ is $ 7.5 $ . Note that the answer is two times the area of the triangle. - Contestant asks a query with $ t = 2 $ , $ i = 1 $ , $ j = 5 $ , $ k = 6 $ . - Jury answers $ -1 $ . The cross product of $ \overrightarrow{A_1A_5} = (2, 2) $ and $ \overrightarrow{A_1A_6} = (4, 1) $ is $ -2 $ . The sign of $ -2 $ is $ -1 $ . - Contestant asks a query with $ t = 2 $ , $ i = 2 $ , $ j = 1 $ , $ k = 4 $ . - Jury answers $ 1 $ . The cross product of $ \overrightarrow{A_2A_1} = (-5, 2) $ and $ \overrightarrow{A_2A_4} = (-2, -1) $ is $ 1 $ . The sign of $ 1 $ is $ 1 $ . - Contestant says that the permutation is $ (1, 3, 4, 2, 6, 5) $ .