Sum of Triangle Area Eighth Powers

You are given an initially empty set of points in the plane. There are N operations. In each operation, a point is inserted into or removed from the set. After each operation, you must compute the following:

If the current set of points is S, consider every triple of distinct points from S that forms a triangle. Let the area of a triangle with vertices \((x_1,y_1)\), \((x_2,y_2)\), and \((x_3,y_3)\) be defined as

[ A = \frac{|x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)|}{2}, ]

For each triangle, take the 8th power of its area, i.e., \(A^8\), and compute the sum over all triangles. Since \(A^8 = \frac{|\text{cross}|^8}{256}\) (where \(|\text{cross}|\) is the absolute value of the triangle's doubled area), the answer can be written as a rational number \(a/b\) in lowest terms.

You are to output the value \(a \cdot b^{-1} \pmod{998244353}\), where \(b^{-1}\) is the modular inverse of \(b\) modulo 998244353.

inputFormat

The first line contains an integer N, the number of operations. Each of the following N lines contains an operation in one of the following formats:

• + x y: Insert the point \((x,y)\) into the set.
• - x y: Remove one occurrence of the point \((x,y)\) from the set.

After each operation, you must output the answer described above.

outputFormat

Output N lines. The i-th line should contain the answer after performing the first i operations. Each answer is computed as \(a \cdot b^{-1} \pmod{998244353}\) as explained in the problem statement.

sample

3
+ 0 0
+ 1 0
+ 0 1

994344961