{"raw_statement":[{"iden":"statement","content":"_Do you know who will fulfill people's wishes on Tanabata?_\n\n_It's $alpha$ Lyra and $alpha$ Aquila._\n\n_By the way, Earth is 25 light-years away from $alpha$ Lyra and 16 light-years away from $alpha$ Aquila. That is to say, it will take 25 or 16 years for the messages sent from Earth to reach these two constellations. That's for granted. Do you understand?_\n\n_– The Boredom of Haruhi Suzumiya_, 2004\n\n   \n\nThis is the $16$th year since the publication of the novel. Apart from the short stories, the author has not updated the series for nine years.\n\nSetsuna is a big fan of the novel. She makes a wish that the author would update as soon as possible, and she will send this message to the stars.\n\nThe universe can be seen as a three-dimensional rectangular coordinate system in which the earth is the origin(i.e., the coordinate of the earth is $(0, 0, 0)$).\n\nSuppose there are $n$ stars, and the coordinate of the $i$-th one is $(x_i, y_i, z_i)$. Any one of them can receive her message.\n\nThe speed of the message is exactly $1$ unit per year in the universe, and the direction can be arbitrary.\n\nSetsuna wants to know the shortest time for the message to reach at least one of the stars after she sends it.\n\nYou can assume that Einstein's theory of relativity doesn't work in Setsuna's world.\n\nThe first line contains one integer $n (1 <= n <= 1000)$, indicating the number of the stars.\n\nThe $i$-th of the next $n$ lines contains three integers $x_i, y_i, z_i (-1000 <= x_i, y_i, z_i <= 1000)$, indicating the coordinate of the $i$-th star.\n\nOutput the shortest time in years, the value should be rounded to three decimal places.\n\nThe first star can receive the message in $sqrt(2)$ years.\n\nThe second and the third star can receive the message in $2$ years.\n\nSo the answer is $sqrt(2)$ years, and it should be rounded to $1. 414$ .\n\n"},{"iden":"input","content":"The first line contains one integer $n (1 <= n <= 1000)$, indicating the number of the stars.The $i$-th of the next $n$ lines contains three integers $x_i, y_i, z_i (-1000 <= x_i, y_i, z_i <= 1000)$, indicating the coordinate of the $i$-th star."},{"iden":"output","content":"Output the shortest time in years, the value should be rounded to three decimal places."},{"iden":"note","content":"The first star can receive the message in $sqrt(2)$ years.The second and the third star can receive the message in $2$ years.So the answer is $sqrt(2)$ years, and it should be rounded to $1. 414$ ."}],"translated_statement":null,"sample_group":[],"show_order":[],"formal_statement":"**Definitions**  \nLet $ S $ be the Special string and $ E $ be the Elegant string, both of length $ n $.  \nLet $ \\text{type}(c) = \\begin{cases} \n0 & \\text{if } c \\text{ is liked by juniors (odd position in alphabet: } a,c,e,\\dots) \\\\\n1 & \\text{if } c \\text{ is liked by seniors (even position in alphabet: } b,d,f,\\dots)\n\\end{cases} $\n\n**Constraints**  \n1. $ |S| = |E| = n $, $ 1 \\leq n \\leq 10^5 $  \n2. $ S, E \\in \\{a,b,\\dots,z\\}^n $  \n\n**Objective**  \nDetermine whether $ S $ can be transformed into $ E $ via any number of adjacent swaps between characters $ c_i, c_j $ such that $ \\text{type}(c_i) \\neq \\text{type}(c_j) $.\n\n**Key Insight**  \nAdjacent swaps between characters of opposite types allow arbitrary reordering *within* each type group. Thus, the multiset of junior-liked characters and the multiset of senior-liked characters must be identical in $ S $ and $ E $.\n\n**Formal Condition**  \nLet $ J_S = \\{ c \\in S \\mid \\text{type}(c) = 0 \\} $, $ \\text{Sen}_S = \\{ c \\in S \\mid \\text{type}(c) = 1 \\} $  \nLet $ J_E = \\{ c \\in E \\mid \\text{type}(c) = 0 \\} $, $ \\text{Sen}_E = \\{ c \\in E \\mid \\text{type}(c) = 1 \\} $\n\nThen:  \n$$\n\\text{Answer} = \\begin{cases}\n\\text{\"Yes\"} & \\text{if } J_S = J_E \\text{ and } \\text{Sen}_S = \\text{Sen}_E \\\\\n\\text{\"No\"} & \\text{otherwise}\n\\end{cases}\n$$","simple_statement":"Given two strings of equal length, determine if you can transform the first string into the second by swapping adjacent characters, but only if one is liked by juniors and the other by seniors. Letters a, c, e, ... (odd positions in alphabet) are liked by juniors; b, d, f, ... (even positions) are liked by seniors. Print \"Yes\" if possible, \"No\" otherwise.","has_page_source":false}