1254C · Point Ordering

Problem: 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.

Note: 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 $(1, 3, 4, 2, 6, 5)$.