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문제

상근이는 하프 마라톤(21km 정도) 대회를 준비하러 동해안으로 떠났다. 상근이의 첫 번째 훈련은 21마일을 뛰는 것이었다.

21마일을 뛰어보니 21킬로미터를 뛴 것보다 더 지치는 것 같았다. 상근이의 친구 정인이는 마라톤은 21마일이 아니고 21킬로미터라고 알려주었다. 또, 21킬로미터는 13마일 같다는 사실도 알려주었다. 21, 13? 상근이는 깊은 깨달음을 얻었다. 두 숫자 모두 피보나치 숫자이다!

피보나치 숫자는 다음과 같이 정의한다.

F1 = 1
F2 = 2
Fn+1 = Fn + Fn-1 (n > 1)

마침 상근이는 훈련을 떠나기 전, 대학에서 피보나치 진법을 배웠다. 모든 양의 정수 X는 서로 다른 피보나치 수의 합으로 나타낼 수 있다. 즉, bk = 1과 bi (1 ≤ i < k) = 0 또는 1이면서 x = ∑i=1..k bi × Fi을 만족하는 정수 k와 b1, b2, ..., bk가 항상 존재한다. 이러한 표현을 b(x) = (bk, bk-1, ..., b1)와 같이 간편하게 쓰기로 한다. 또, 여러가지 표현이 생기는 것을 막기 위해 bi × bi-1 = 0이어야 한다. (i > 1)

예를 들어, 21은 (1, 0, 0, 0, 0, 0, 0)로, 13은 (1, 0, 0, 0, 0, 0)으로 나타낼 수 있다. 상근이는 이런 피보나치 진법을 이용해서 x킬로미터를 y마일로 바꾸는 방법을 만들었다.

먼저, x를 피보나치 진법 b(x)로 나타낸다. 그 다음, b(x)를 오른쪽으로 한 비트씩 시프트 시킨다. (맨 마지막 비트는 버린다) 이것을 b(y)라고 한다. 이제 b(y)를 이용해서 y를 계산하면 된다.

예를 들어, 42는 피보나치 진법으로 (1, 0, 0, 1, 0, 0, 0, 0)이다. 이걸 오른쪽으로 한 비트씩 시프트 시키면 (1, 0, 0, 1, 0, 0, 0)이 되고, 0×1 + 0×2 + 0×3 + 1×5 + 0×8 + 0×13 + 1×21 = 26이 된다.

킬로미터 값이 주어졌을 때, 상근이의 알고리즘을 이용해서 마일로 바꾸는 프로그램을 작성하시오.

입력

첫째 줄에 마일로 바꾸려고 하는 킬로미터 값의 수 t가 주어진다. (0 < t < 25000) 다음 t개 줄에는 마일로 바꿀 킬로미터 값 x가 주어진다. (2 × x < 25000)

출력

각각의 킬로미터 거리 x에 대해서, 상근이의 알고리즘을 이용해 마일로 바꾼 값 y를 출력한다.

예제 입력 1

5
42
100
180
300
360

예제 출력 1

26
62
111
185
222

힌트

[{"problem_id":"6504","problem_lang":"0","title":"\ud0ac\ub85c\ubbf8\ud130\ub97c \ub9c8\uc77c\ub85c","description":"<p>\uc0c1\uadfc\uc774\ub294 \ud558\ud504 \ub9c8\ub77c\ud1a4(21km \uc815\ub3c4) \ub300\ud68c\ub97c \uc900\ube44\ud558\ub7ec \ub3d9\ud574\uc548\uc73c\ub85c \ub5a0\ub0ac\ub2e4. \uc0c1\uadfc\uc774\uc758 \uccab \ubc88\uc9f8 \ud6c8\ub828\uc740 21\ub9c8\uc77c\uc744 \ub6f0\ub294 \uac83\uc774\uc5c8\ub2e4.<\/p>\r\n\r\n<p>21\ub9c8\uc77c\uc744 \ub6f0\uc5b4\ubcf4\ub2c8 21\ud0ac\ub85c\ubbf8\ud130\ub97c \ub6f4 \uac83\ubcf4\ub2e4 \ub354 \uc9c0\uce58\ub294 \uac83 \uac19\uc558\ub2e4. \uc0c1\uadfc\uc774\uc758 \uce5c\uad6c \uc815\uc778\uc774\ub294 \ub9c8\ub77c\ud1a4\uc740 21\ub9c8\uc77c\uc774 \uc544\ub2c8\uace0 21\ud0ac\ub85c\ubbf8\ud130\ub77c\uace0 \uc54c\ub824\uc8fc\uc5c8\ub2e4. \ub610, 21\ud0ac\ub85c\ubbf8\ud130\ub294 13\ub9c8\uc77c \uac19\ub2e4\ub294 \uc0ac\uc2e4\ub3c4 \uc54c\ub824\uc8fc\uc5c8\ub2e4. 21, 13? \uc0c1\uadfc\uc774\ub294 \uae4a\uc740 \uae68\ub2ec\uc74c\uc744 \uc5bb\uc5c8\ub2e4. \ub450 \uc22b\uc790 \ubaa8\ub450 \ud53c\ubcf4\ub098\uce58 \uc22b\uc790\uc774\ub2e4!<\/p>\r\n\r\n<p>\ud53c\ubcf4\ub098\uce58 \uc22b\uc790\ub294 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<\/p>\r\n\r\n<p>F<sub>1<\/sub> = 1<br \/>\r\nF<sub>2<\/sub> = 2<br \/>\r\nF<sub>n+1<\/sub> = F<sub>n<\/sub> + F<sub>n-1<\/sub> (n &gt; 1)<\/p>\r\n\r\n<p>\ub9c8\uce68 \uc0c1\uadfc\uc774\ub294 \ud6c8\ub828\uc744 \ub5a0\ub098\uae30 \uc804, \ub300\ud559\uc5d0\uc11c \ud53c\ubcf4\ub098\uce58 \uc9c4\ubc95\uc744 \ubc30\uc6e0\ub2e4. \ubaa8\ub4e0 \uc591\uc758 \uc815\uc218 X\ub294 \uc11c\ub85c \ub2e4\ub978 \ud53c\ubcf4\ub098\uce58 \uc218\uc758 \ud569\uc73c\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4. \uc989, b<sub>k<\/sub> = 1\uacfc b<sub>i<\/sub> (1 &le; i &lt; k) = 0 \ub610\ub294 1\uc774\uba74\uc11c x = &sum;i=<sub>1..k<\/sub> b<sub>i<\/sub> &times; F<sub>i<\/sub>\uc744 \ub9cc\uc871\ud558\ub294 \uc815\uc218 k\uc640 b<sub>1<\/sub>, b<sub>2<\/sub>, ..., b<sub>k<\/sub>\uac00 \ud56d\uc0c1 \uc874\uc7ac\ud55c\ub2e4. \uc774\ub7ec\ud55c \ud45c\ud604\uc744 b(x) = (b<sub>k<\/sub>, b<sub>k-1<\/sub>, ..., b<sub>1<\/sub>)\uc640 \uac19\uc774 \uac04\ud3b8\ud558\uac8c \uc4f0\uae30\ub85c \ud55c\ub2e4. \ub610, \uc5ec\ub7ec\uac00\uc9c0 \ud45c\ud604\uc774 \uc0dd\uae30\ub294 \uac83\uc744 \ub9c9\uae30 \uc704\ud574 b<sub>i<\/sub> &times; b<sub>i-1<\/sub> = 0\uc774\uc5b4\uc57c \ud55c\ub2e4. (i &gt; 1)<\/p>\r\n\r\n<p>\uc608\ub97c \ub4e4\uc5b4, 21\uc740 (1, 0, 0, 0, 0, 0, 0)\ub85c, 13\uc740 (1, 0, 0, 0, 0, 0)\uc73c\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4. \uc0c1\uadfc\uc774\ub294 \uc774\ub7f0 \ud53c\ubcf4\ub098\uce58 \uc9c4\ubc95\uc744 \uc774\uc6a9\ud574\uc11c x\ud0ac\ub85c\ubbf8\ud130\ub97c y\ub9c8\uc77c\ub85c \ubc14\uafb8\ub294 \ubc29\ubc95\uc744 \ub9cc\ub4e4\uc5c8\ub2e4.<\/p>\r\n\r\n<p>\uba3c\uc800, x\ub97c \ud53c\ubcf4\ub098\uce58 \uc9c4\ubc95 b(x)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \uadf8 \ub2e4\uc74c, b(x)\ub97c \uc624\ub978\ucabd\uc73c\ub85c \ud55c \ube44\ud2b8\uc529 \uc2dc\ud504\ud2b8 \uc2dc\ud0a8\ub2e4. (\ub9e8 \ub9c8\uc9c0\ub9c9 \ube44\ud2b8\ub294 \ubc84\ub9b0\ub2e4) \uc774\uac83\uc744 b(y)\ub77c\uace0 \ud55c\ub2e4. \uc774\uc81c b(y)\ub97c \uc774\uc6a9\ud574\uc11c y\ub97c \uacc4\uc0b0\ud558\uba74 \ub41c\ub2e4.<\/p>\r\n\r\n<p>\uc608\ub97c \ub4e4\uc5b4, 42\ub294 \ud53c\ubcf4\ub098\uce58 \uc9c4\ubc95\uc73c\ub85c (1, 0, 0, 1, 0, 0, 0, 0)\uc774\ub2e4. \uc774\uac78 \uc624\ub978\ucabd\uc73c\ub85c \ud55c \ube44\ud2b8\uc529 \uc2dc\ud504\ud2b8 \uc2dc\ud0a4\uba74 (1, 0, 0, 1, 0, 0, 0)\uc774 \ub418\uace0, 0&times;1 + 0&times;2 + 0&times;3 + 1&times;5 + 0&times;8 + 0&times;13 + 1&times;21 = 26\uc774 \ub41c\ub2e4.<\/p>\r\n\r\n<p>\ud0ac\ub85c\ubbf8\ud130 \uac12\uc774 \uc8fc\uc5b4\uc84c\uc744 \ub54c, \uc0c1\uadfc\uc774\uc758 \uc54c\uace0\ub9ac\uc998\uc744 \uc774\uc6a9\ud574\uc11c \ub9c8\uc77c\ub85c \ubc14\uafb8\ub294 \ud504\ub85c\uadf8\ub7a8\uc744 \uc791\uc131\ud558\uc2dc\uc624.<\/p>\r\n","input":"<p>\uccab\uc9f8 \uc904\uc5d0 \ub9c8\uc77c\ub85c \ubc14\uafb8\ub824\uace0 \ud558\ub294 \ud0ac\ub85c\ubbf8\ud130 \uac12\uc758 \uc218 t\uac00 \uc8fc\uc5b4\uc9c4\ub2e4. (0 &lt; t &lt; 25000) \ub2e4\uc74c t\uac1c \uc904\uc5d0\ub294 \ub9c8\uc77c\ub85c \ubc14\uafc0 \ud0ac\ub85c\ubbf8\ud130 \uac12 x\uac00 \uc8fc\uc5b4\uc9c4\ub2e4. (2 &times; x &lt; 25000)<\/p>\r\n","output":"<p>\uac01\uac01\uc758 \ud0ac\ub85c\ubbf8\ud130 \uac70\ub9ac x\uc5d0 \ub300\ud574\uc11c, \uc0c1\uadfc\uc774\uc758 \uc54c\uace0\ub9ac\uc998\uc744 \uc774\uc6a9\ud574 \ub9c8\uc77c\ub85c \ubc14\uafbc \uac12 y\ub97c \ucd9c\ub825\ud55c\ub2e4.<\/p>\r\n","hint":"","original":"0","problem_lang_code":"\ud55c\uad6d\uc5b4"},{"problem_id":"6504","problem_lang":"1","title":"Convert Kilometers to Miles","description":"<p>This year, Bruce Force spends his vacation in Flagstaff, Arizona, where he wants to practice for his next half marathon (a race over 21 km). At his first training he runs to his friend Greedy Gonzales&#39; home which is 21 miles away from Flagstaff.<\/p>\r\n\r\n<p>Arriving there, he is very tired and realizes that 21 miles are much more than 21 km. Greedy Gonzales tells him that 21 km equals 13 miles. 21, 13? Bruce realizes immediately that there must be a deeper relation! Both, 13 and 21 are Fibonacci numbers!<\/p>\r\n\r\n<p>Fibonacci numbers can be defined as follows:<\/p>\r\n\r\n<p>F<sub>1<\/sub> = 1<br \/>\r\nF<sub>2<\/sub> = 2<br \/>\r\nF<sub>n+1<\/sub> = F<sub>n<\/sub>+F<sub>n-1<\/sub> for n&gt;1<\/p>\r\n\r\n<p>Bruce has just learned about the Fibonacci number system at his university. Each positive integer x can be written as the sum of different Fibonacci numbers, so this means that there exists numbers k and b<sub>1<\/sub>, b<sub>2<\/sub>, ..., b<sub>k<\/sub> such that x = &sum;<sub>i=1..k<\/sub> b<sub>i<\/sub> * F<sub>i<\/sub>, where b<sub>k<\/sub> = 1 and b<sub>i<\/sub> (1 &le; i &lt; k) is either 0 or 1. In short, we can write the representation as: b(x) = (b<sub>k<\/sub>, b<sub>k-1<\/sub>, ..., b<sub>1<\/sub>). To make the representation unique, we require that b<sub>i<\/sub> * b<sub>i-1<\/sub> = 0 for all i &gt; 1.<\/p>\r\n\r\n<p>For example 21 can be represented as (1,0,0,0,0,0,0) and 13 as (1,0,0,0,0,0) in the Fibonacci system. Bruce notices that one can convert a distance x in kilometers into a corresponding distance y to miles as follows: First, write down x in its Fibonacci system representation b(x). Second, shift the bits of b(x) one position to the right (the last bit is deleted) and get b(y). Third, calculate y from b(y) by evaluating the sum given above.<\/p>\r\n\r\n<p>For example, the number 42 written in the Fibonacci system is (1,0,0,1,0,0,0,0). In step two we would shift the bits one position to the right and get (1,0,0,1,0,0,0). In the third step, we would calculate 0*1 + 0*2 + 0*3 + 1*5 + 0*8 + 0*13 + 1*21 = 26.<\/p>\r\n\r\n<p>Now it&#39;s your turn to write a program for Bruce that converts kilometers into miles according to Bruce&#39;s algorithm.<\/p>\r\n","input":"<p>The first line of the input contains t, the number of distances Bruce wants to convert from kilometers to miles (0&lt;t&lt;25000). Each of the next t lines contains an integer distance x (2 &lt; x &lt; 25000) in kilometers.<\/p>\r\n","output":"<p>For each distance x in kilometers output the distance y in miles calculated according to Bruce&#39;s algorithm.<\/p>\r\n","hint":"","original":"1","problem_lang_code":"\uc601\uc5b4"}]