{"raw_statement":[{"iden":"problem statement","content":"You are given $N$ points $(x_i,y_i)$ located on a two-dimensional plane. Consider a subset $S$ of the $N$ points that forms a convex polygon. Here, we say a set of points $S$ _forms a convex polygon_ when there exists a convex polygon with a positive area that has the same set of vertices as $S$. All the interior angles of the polygon must be strictly less than $180°$.\n\n![image](https://atcoder.jp/img/arc082/cddb0c267926c2add885ca153c47ad8a.png)\n\nFor example, in the figure above, {$A,C,E$} and {$B,D,E$} form convex polygons; {$A,C,D,E$}, {$A,B,C,E$}, {$A,B,C$}, {$D,E$} and {} do not.\nFor a given set $S$, let $n$ be the number of the points among the $N$ points that are inside the convex hull of $S$ (including the boundary and vertices). Then, we will define the _score_ of $S$ as $2^{n-|S|}$.\nCompute the scores of all possible sets $S$ that form convex polygons, and find the sum of all those scores.\nHowever, since the sum can be extremely large, print the sum modulo $998244353$."},{"iden":"constraints","content":"*   $1≤N≤200$\n*   $0≤x_i,y_i<10^4 (1≤i≤N)$\n*   If $i≠j$, $x_i≠x_j$ or $y_i≠y_j$.\n*   $x_i$ and $y_i$ are integers."},{"iden":"input","content":"The input is given from Standard Input in the following format:\n\n$N$\n$x_1$ $y_1$\n$x_2$ $y_2$\n$:$\n$x_N$ $y_N$"},{"iden":"sample input 1","content":"4\n0 0\n0 1\n1 0\n1 1"},{"iden":"sample output 1","content":"5\n\nWe have five possible sets as $S$, four sets that form triangles and one set that forms a square. Each of them has a score of $2^0=1$, so the answer is $5$."},{"iden":"sample input 2","content":"5\n0 0\n0 1\n0 2\n0 3\n1 1"},{"iden":"sample output 2","content":"11\n\nWe have three \"triangles\" with a score of $1$ each, two \"triangles\" with a score of $2$ each, and one \"triangle\" with a score of $4$. Thus, the answer is $11$."},{"iden":"sample input 3","content":"1\n3141 2718"},{"iden":"sample output 3","content":"0\n\nThere are no possible set as $S$, so the answer is $0$."}],"translated_statement":null,"sample_group":[],"show_order":["default"],"formal_statement":null,"simple_statement":null,"has_page_source":true}