{"trustable":true,"prependHtml":"\u003cstyle type\u003d\u0027text/css\u0027\u003e\n .input, .output {\n border: 1px solid #888888;\n }\n .output {\n margin-bottom: 1em;\n position: relative;\n top: -1px;\n }\n .output pre, .input pre {\n background-color: #EFEFEF;\n line-height: 1.25em;\n margin: 0;\n padding: 0.25em;\n }\n \u003c/style\u003e\n \u003clink rel\u003d\"stylesheet\" href\u003d\"//codeforces.org/s/96598/css/problem-statement.css\" type\u003d\"text/css\" /\u003e\u003cscript\u003e window.katexOptions \u003d { disable: true }; \u003c/script\u003e\n\u003cscript type\u003d\"text/x-mathjax-config\"\u003e\n MathJax.Hub.Config({\n tex2jax: {\n inlineMath: [[\u0027$$$\u0027,\u0027$$$\u0027], [\u0027$\u0027,\u0027$\u0027]],\n displayMath: [[\u0027$$$$$$\u0027,\u0027$$$$$$\u0027], [\u0027$$\u0027,\u0027$$\u0027]]\n }\n });\n\u003c/script\u003e\n\u003cscript type\u003d\"text/javascript\" async src\u003d\"https://mathjax.codeforces.org/MathJax.js?config\u003dTeX-AMS_HTML-full\"\u003e\u003c/script\u003e","sections":[{"title":"","value":{"format":"HTML","content":"\u003cp\u003eConsider an initial set \u003cspan class\u003d\"tex-span\"\u003e\u003ci\u003eA\u003c/i\u003e\u003c/span\u003e consisting of \u003cspan class\u003d\"tex-span\"\u003e\u003ci\u003eN\u003c/i\u003e\u003c/span\u003e points in three-dimensional space. Each point is associated with a real value \u003cspan class\u003d\"tex-span\"\u003e\u003ci\u003ep\u003c/i\u003e\u003csub class\u003d\"lower-index\"\u003e\u003ci\u003ei\u003c/i\u003e\u003c/sub\u003e\u003c/span\u003e that is equal to the probability that \u003cspan class\u003d\"tex-span\"\u003e\u003ci\u003ei\u003c/i\u003e\u003c/span\u003e-th point is included in the final set \u003cspan class\u003d\"tex-span\"\u003e\u003ci\u003eB\u003c/i\u003e\u003c/span\u003e. Your task is to calculate the expected number of \u003cspan class\u003d\"tex-font-style-it\"\u003eborder points\u003c/span\u003e for convex hull of \u003cspan class\u003d\"tex-span\"\u003e\u003ci\u003eB\u003c/i\u003e\u003c/span\u003e.\u003c/p\u003e\u003cp\u003eA point \u003cimg align\u003d\"middle\" class\u003d\"tex-formula\" src\u003d\"CDN_BASE_URL/e40a9cda60af863c92b001738139ff27?v\u003d1714449208\" style\u003d\"max-width: 100.0%;max-height: 100.0%;\"\u003e is called a \u003cspan class\u003d\"tex-font-style-it\"\u003eborder point\u003c/span\u003e for the convex hull of set \u003cspan class\u003d\"tex-span\"\u003e\u003ci\u003eB\u003c/i\u003e\u003c/span\u003e if there is no way to choose points \u003cimg align\u003d\"middle\" class\u003d\"tex-formula\" src\u003d\"CDN_BASE_URL/f5ba99033412de14675b919ea8eed60a?v\u003d1714449208\" style\u003d\"max-width: 100.0%;max-height: 100.0%;\"\u003e such that \u003cspan class\u003d\"tex-span\"\u003e\u003ci\u003ep\u003c/i\u003e\u003c/span\u003e lies strictly inside the tetrahedron \u003cspan class\u003d\"tex-span\"\u003e\u003ci\u003eabcd\u003c/i\u003e\u003c/span\u003e.\u003c/p\u003e"}},{"title":"Input","value":{"format":"HTML","content":"\u003cp\u003eThe first line contains the number of points \u003cspan class\u003d\"tex-span\"\u003e\u003ci\u003eN\u003c/i\u003e\u003c/span\u003e (\u003cspan class\u003d\"tex-span\"\u003e1 ≤ \u003ci\u003eN\u003c/i\u003e ≤ 50\u003c/span\u003e). The next \u003cspan class\u003d\"tex-span\"\u003e\u003ci\u003eN\u003c/i\u003e\u003c/span\u003e lines describe points of \u003cspan class\u003d\"tex-span\"\u003e\u003ci\u003eA\u003c/i\u003e\u003c/span\u003e in the form \u003cspan class\u003d\"tex-span\"\u003e\u003ci\u003ep\u003c/i\u003e\u003csub class\u003d\"lower-index\"\u003e\u003ci\u003ei\u003c/i\u003e\u003c/sub\u003e\u0026nbsp;\u003ci\u003ex\u003c/i\u003e\u003csub class\u003d\"lower-index\"\u003e\u003ci\u003ei\u003c/i\u003e\u003c/sub\u003e\u0026nbsp;\u003ci\u003ey\u003c/i\u003e\u003csub class\u003d\"lower-index\"\u003e\u003ci\u003ei\u003c/i\u003e\u003c/sub\u003e\u0026nbsp;\u003ci\u003ez\u003c/i\u003e\u003csub class\u003d\"lower-index\"\u003e\u003ci\u003ei\u003c/i\u003e\u003c/sub\u003e\u003c/span\u003e, which are the probability of the \u003cspan class\u003d\"tex-span\"\u003e\u003ci\u003ei\u003c/i\u003e\u003c/span\u003e-th point to be included in the final set and the coordinates of the \u003cspan class\u003d\"tex-span\"\u003e\u003ci\u003ei\u003c/i\u003e\u003c/span\u003e-th point (\u003cspan class\u003d\"tex-span\"\u003e0.00 ≤ \u003ci\u003ep\u003c/i\u003e\u003csub class\u003d\"lower-index\"\u003e\u003ci\u003ei\u003c/i\u003e\u003c/sub\u003e ≤ 1.00\u003c/span\u003e, \u003cspan class\u003d\"tex-span\"\u003e - 1000 ≤ \u003ci\u003ex\u003c/i\u003e\u003csub class\u003d\"lower-index\"\u003e\u003ci\u003ei\u003c/i\u003e\u003c/sub\u003e, \u003ci\u003ey\u003c/i\u003e\u003csub class\u003d\"lower-index\"\u003e\u003ci\u003ei\u003c/i\u003e\u003c/sub\u003e, \u003ci\u003ez\u003c/i\u003e\u003csub class\u003d\"lower-index\"\u003e\u003ci\u003ei\u003c/i\u003e\u003c/sub\u003e ≤ 1000\u003c/span\u003e). Probabilities are given with exactly two digits after the decimal point.\u003c/p\u003e\u003cp\u003eIt is guaranteed that no two points of \u003cspan class\u003d\"tex-span\"\u003e\u003ci\u003eA\u003c/i\u003e\u003c/span\u003e coincide, no three points of \u003cspan class\u003d\"tex-span\"\u003e\u003ci\u003eA\u003c/i\u003e\u003c/span\u003e are collinear and no four points of \u003cspan class\u003d\"tex-span\"\u003e\u003ci\u003eA\u003c/i\u003e\u003c/span\u003e are coplanar.\u003c/p\u003e"}},{"title":"Output","value":{"format":"HTML","content":"\u003cp\u003eOutput the expected number of \u003cspan class\u003d\"tex-font-style-it\"\u003eborder points\u003c/span\u003e. Your answer will be considered correct if its absolute error is less then \u003cspan class\u003d\"tex-span\"\u003e10\u003csup class\u003d\"upper-index\"\u003e - 6\u003c/sup\u003e\u003c/span\u003e.\u003c/p\u003e"}},{"title":"Examples","value":{"format":"HTML","content":"\u003ctable class\u003d\u0027vjudge_sample\u0027\u003e\n\u003cthead\u003e\n \u003ctr\u003e\n \u003cth\u003eInput\u003c/th\u003e\n \u003cth\u003eOutput\u003c/th\u003e\n \u003c/tr\u003e\n\u003c/thead\u003e\n\u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd\u003e\u003cpre\u003e5\n0.20 0 0 0\n0.40 0 0 4\n0.60 0 4 0\n0.80 4 0 0\n0.50 1 1 1\n\u003c/pre\u003e\u003c/td\u003e\n \u003ctd\u003e\u003cpre\u003e2.4808000000\n\u003c/pre\u003e\u003c/td\u003e\n \u003c/tr\u003e\n\u003c/tbody\u003e\n\u003c/table\u003e\n"}}]}