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문제

피보나치 수열의 매력에 빠진 현욱이는 피보나치 수열을 너무나도 사랑한 나머지 '피보나치 문제해결전략' 책을 사서 읽다가 Gabonacci 수열의 존재에 대해 알게 된다.

당연히 피보나치 수열에 대해서 알고 있을 것이다. 만약 모른다면  2747번 문제를 풀어 보는 것을 권장한다. n번째 피보나치 항을 Fn이라 할 때 F1 = 1, F2 = 2이며, 그 이후의 항들은 모두 바로 전 2개 항의 합이다. 이러한 귀납적 정의에 따라 피보나치 수열은 1, 1, 2, 3, 5, 8, 13, . . .. 으로 이어진다.

이제 이를 좀 더 일반화해 보자. 어떤 수열이 피보나치 수열과 같은 재귀적 정의  

    G i = Gi-1 + Gi-2 for i > 2

를 따르지만, 처음 두 항을 G 1 ≤ G2 를 만족하게 하면서 임의로 설정할 것이다. 이를 Gabonacci 수열이라고 한다. 만약 G1 = 1, G2 = 3이면, 널리 알려진 수열인 Lucas numbers(1, 3, 4, 7, 11, 18, 29, . . ..)를 얻을 수 있다.

적절한 두 개의 첫 항을 골라서, 당신이 원하는 어떤 자연수를 반드시 Gabonacci 수열에서 등장하게 할 수 있다. 예를 들면, n은 1, n-1로 시작하는 Gabonacci 수열에서 등장한다. 그러나 현욱이는 이건 너무 쉽다고 생각했다. 그래서 가능한 한 작은 항으로 시작해 n이 등장하게 하고 싶다.

입력

첫 번째 줄에는 테스트 케이스의 개수 T가 주어진다. (T ≤ 100) 두 번째 줄부터 각 테스트 케이스에 대해 나타나게 해야 할 정수 n이 주어진다. (2 ≤ n ≤ 109)

출력

각 테스트 케이스마다 한 줄에 두 자연수 a, b를 출력한다. (0 < a ≤ b) G1 = a, G2 = b, 그리고 어떤 자연수 k에 대해 Gk = n 이다. a, b는 가능한 한 제일 작아야 하며, 이는 어떤 자연수 a', b'에 대해서 a', b'로 시작하는 Gabonacci 수열에서 n이 등장하고, b' < b 이거나 b' = b 이고 a' < a 인 a', b' 가 존재하면 안 된다는 뜻이다.

예제 입력 1

5
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1000
1573655
842831057

예제 출력 1

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2 7
[{"problem_id":"10327","problem_lang":"0","title":"\ud53c\ubcf4\ub098\uce58 \ubb38\uc81c\ud574\uacb0\uc804\ub7b5","description":"<p><img alt=\"\" src=\"\/upload\/images2\/crime_scene.png\" style=\"float:right; height:425px; width:286px\" \/><\/p>\r\n\r\n<p>\ud53c\ubcf4\ub098\uce58 \uc218\uc5f4\uc758 \ub9e4\ub825\uc5d0 \ube60\uc9c4 \ud604\uc6b1\uc774\ub294 \ud53c\ubcf4\ub098\uce58 \uc218\uc5f4\uc744 \ub108\ubb34\ub098\ub3c4 \uc0ac\ub791\ud55c \ub098\uba38\uc9c0 &#39;\ud53c\ubcf4\ub098\uce58 \ubb38\uc81c\ud574\uacb0\uc804\ub7b5&#39; \ucc45\uc744 \uc0ac\uc11c \uc77d\ub2e4\uac00 Gabonacci \uc218\uc5f4\uc758 \uc874\uc7ac\uc5d0 \ub300\ud574 \uc54c\uac8c \ub41c\ub2e4.<\/p>\r\n\r\n<div>\r\n<p>\ub2f9\uc5f0\ud788 \ud53c\ubcf4\ub098\uce58 \uc218\uc5f4\uc5d0 \ub300\ud574\uc11c \uc54c\uace0 \uc788\uc744 \uac83\uc774\ub2e4. \ub9cc\uc57d \ubaa8\ub978\ub2e4\uba74&nbsp; <a href=\"https:\/\/www.acmicpc.net\/problem\/2747\">2747\ubc88<\/a> \ubb38\uc81c\ub97c \ud480\uc5b4 \ubcf4\ub294 \uac83\uc744 \uad8c\uc7a5\ud55c\ub2e4. n\ubc88\uc9f8 \ud53c\ubcf4\ub098\uce58 \ud56d\uc744 F<sub>n<\/sub>\uc774\ub77c \ud560 \ub54c F<sub>1<\/sub> = 1, F<sub>2<\/sub> = 2\uc774\uba70, \uadf8 \uc774\ud6c4\uc758 \ud56d\ub4e4\uc740 \ubaa8\ub450 \ubc14\ub85c \uc804 2\uac1c \ud56d\uc758 \ud569\uc774\ub2e4. \uc774\ub7ec\ud55c \uadc0\ub0a9\uc801 \uc815\uc758\uc5d0 \ub530\ub77c \ud53c\ubcf4\ub098\uce58 \uc218\uc5f4\uc740 1, 1, 2, 3, 5, 8, 13, . . .. \uc73c\ub85c \uc774\uc5b4\uc9c4\ub2e4.<\/p>\r\n\r\n<p>\uc774\uc81c \uc774\ub97c \uc880 \ub354 \uc77c\ubc18\ud654\ud574 \ubcf4\uc790. \uc5b4\ub5a4 \uc218\uc5f4\uc774&nbsp;\ud53c\ubcf4\ub098\uce58 \uc218\uc5f4\uacfc \uac19\uc740&nbsp;\uc7ac\uadc0\uc801 \uc815\uc758 &nbsp;<\/p>\r\n\r\n<p>&nbsp;&nbsp;&nbsp;&nbsp;G <sub>i<\/sub> = G<sub>i-1<\/sub> + G<sub>i-2<\/sub> for i &gt; 2<\/p>\r\n\r\n<p>\ub97c \ub530\ub974\uc9c0\ub9cc, \ucc98\uc74c \ub450 \ud56d\uc744&nbsp;G <sub>1<\/sub> &le; G<sub>2<\/sub>&nbsp;\ub97c \ub9cc\uc871\ud558\uac8c \ud558\uba74\uc11c&nbsp;\uc784\uc758\ub85c \uc124\uc815\ud560 \uac83\uc774\ub2e4. \uc774\ub97c Gabonacci \uc218\uc5f4\uc774\ub77c\uace0 \ud55c\ub2e4. \ub9cc\uc57d&nbsp;G<sub>1<\/sub> = 1, G<sub>2<\/sub> = 3\uc774\uba74, \ub110\ub9ac \uc54c\ub824\uc9c4 \uc218\uc5f4\uc778 Lucas numbers(1, 3, 4, 7, 11, 18, 29, . . ..)\ub97c \uc5bb\uc744 \uc218 \uc788\ub2e4.<\/p>\r\n\r\n<p>\uc801\uc808\ud55c \ub450 \uac1c\uc758 \uccab \ud56d\uc744 \uace8\ub77c\uc11c, \ub2f9\uc2e0\uc774 \uc6d0\ud558\ub294 \uc5b4\ub5a4 \uc790\uc5f0\uc218\ub97c&nbsp;\ubc18\ub4dc\uc2dc Gabonacci \uc218\uc5f4\uc5d0\uc11c \ub4f1\uc7a5\ud558\uac8c \ud560 \uc218 \uc788\ub2e4. \uc608\ub97c \ub4e4\uba74, n\uc740 1, n-1\ub85c \uc2dc\uc791\ud558\ub294 Gabonacci \uc218\uc5f4\uc5d0\uc11c \ub4f1\uc7a5\ud55c\ub2e4. \uadf8\ub7ec\ub098 \ud604\uc6b1\uc774\ub294&nbsp;\uc774\uac74 \ub108\ubb34 \uc27d\ub2e4\uace0 \uc0dd\uac01\ud588\ub2e4. \uadf8\ub798\uc11c \uac00\ub2a5\ud55c \ud55c \uc791\uc740 \ud56d\uc73c\ub85c \uc2dc\uc791\ud574 n\uc774 \ub4f1\uc7a5\ud558\uac8c \ud558\uace0 \uc2f6\ub2e4.<\/p>\r\n<\/div>\r\n","input":"<p>\uccab \ubc88\uc9f8 \uc904\uc5d0\ub294 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\uc758 \uac1c\uc218 T\uac00 \uc8fc\uc5b4\uc9c4\ub2e4. (T &le; 100) \ub450 \ubc88\uc9f8 \uc904\ubd80\ud130 \uac01 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\uc5d0 \ub300\ud574 \ub098\ud0c0\ub098\uac8c \ud574\uc57c \ud560&nbsp;\uc815\uc218 n\uc774 \uc8fc\uc5b4\uc9c4\ub2e4.&nbsp;(2 &le; n &le; 10<sup>9<\/sup>)<\/p>\r\n","output":"<p>\uac01 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\ub9c8\ub2e4 \ud55c \uc904\uc5d0 \ub450 \uc790\uc5f0\uc218 a, b\ub97c \ucd9c\ub825\ud55c\ub2e4. (0&nbsp;&lt;&nbsp;a &le; b) G<sub>1<\/sub> = a, G<sub>2<\/sub> = b, \uadf8\ub9ac\uace0 \uc5b4\ub5a4 \uc790\uc5f0\uc218 k\uc5d0 \ub300\ud574 G<sub>k<\/sub> = n \uc774\ub2e4. a, b\ub294 \uac00\ub2a5\ud55c \ud55c \uc81c\uc77c \uc791\uc544\uc57c \ud558\uba70, \uc774\ub294&nbsp;\uc5b4\ub5a4 \uc790\uc5f0\uc218 a&#39;, b&#39;\uc5d0 \ub300\ud574\uc11c a&#39;, b&#39;\ub85c \uc2dc\uc791\ud558\ub294 Gabonacci \uc218\uc5f4\uc5d0\uc11c n\uc774 \ub4f1\uc7a5\ud558\uace0,&nbsp;b&#39; &lt; b \uc774\uac70\ub098 b&#39; = b \uc774\uace0 a&#39; &lt; a \uc778 a&#39;, b&#39; \uac00 \uc874\uc7ac\ud558\uba74 \uc548 \ub41c\ub2e4\ub294 \ub73b\uc774\ub2e4.<\/p>\r\n","hint":"","original":"0","problem_lang_code":"\ud55c\uad6d\uc5b4"},{"problem_id":"10327","problem_lang":"1","title":"Interesting Integers","description":"<p><a href=\"https:\/\/xkcd.com\/587\/\"><img alt=\"\" src=\"\/upload\/images2\/crime_scene.png\" style=\"float:right; height:425px; width:286px\" \/><\/a>Undoubtedly you know of the Fibonacci numbers. Starting with F<sub>1<\/sub> = 1 and F<sub>2<\/sub> = 1, every next number is the sum of the two previous ones. This results in the sequence 1, 1, 2, 3, 5, 8, 13, . . ..<\/p>\r\n\r\n<p>Now let us consider more generally sequences that obey the same recursion relation<\/p>\r\n\r\n<p style=\"text-align:center\">G<sub>i<\/sub> = G<sub>i-1<\/sub> + G<sub>i-2<\/sub> for i &gt; 2<\/p>\r\n\r\n<p>but start with two numbers G<sub>1<\/sub> &le; G<sub>2<\/sub> of our own choice. We shall call these Gabonacci sequences. For example, if one uses G<sub>1<\/sub> = 1 and G<sub>2<\/sub> = 3, one gets what are known as the Lucas numbers: 1, 3, 4, 7, 11, 18, 29, . . .. These numbers are &ndash; apart from 1 and 3 &ndash; different from the Fibonacci numbers.<\/p>\r\n\r\n<p>By choosing the first two numbers appropriately, you can get any number you like to appear in the Gabonacci sequence. For example, the number n appears in the sequence that starts with 1 and n &minus; 1, but that is a bit lame. It would be more fun to start with numbers that are as small as possible, would you not agree?<\/p>\r\n","input":"<p>On the first line one positive number: the number of test cases, at most 100. After that per test case:<\/p>\r\n\r\n<ul>\r\n\t<li>one line with a single integer n (2 &le; n &le; 10<sup>9<\/sup>): the number to appear in the sequence.<\/li>\r\n<\/ul>\r\n","output":"<p>Per test case:<\/p>\r\n\r\n<ul>\r\n\t<li>one line with two integers a and b (0 &lt; a &le; b), such that, for G<sub>1<\/sub> = a and G<sub>2<\/sub> = b, G<sub>k<\/sub> = n for some k. These numbers should be the smallest possible, i.e., there should be no numbers a&#39; and b&#39; with the same property, for which b&#39; &lt; b, or for which b&#39; = b and a&#39; &lt; a.<\/li>\r\n<\/ul>\r\n","hint":"","original":"1","problem_lang_code":"\uc601\uc5b4"}]