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문제

보통 고등학교 화학 시간에는 화학 반응식을 배운다. 이 식은 다음과 같이 생겼다.

(1) 2H2 + O2 → 2H2O
(2) Ca(OH)2 + CO2 → CaCO3 + H2O
(3) N2 + 3H2 → 2NH3

위의 식 (1)~(3)은 모두 좌변과 우변의 균형이 맞추어져 있다.  하지만, 아래 반응식은 그렇지 않다.

(4) Al + O2 → Al2O3
(5) C3H8 + O2 → CO2 + H2O

모든 화학 반응식은 질량 보존의 법칙을 지켜야 한다. 즉, 각 원소(H, O, Ca, Al, ...)의 개수는 화학 반응으로 변하지 않아야 한다. 따라서, 좌변과 우변에 분자의 개수를 맞추어 주어아 한다.

(6) 4Al + 3O2 → 2Al2O3
(7) C3H8 + 5O2 → 3CO2 + 4H2O

반응식 (6)의 계수는 왼쪽부터 오른쪽까지 (4,3,2)이고, 반응식 (7)의 계수는 (1,5,3,4)이다. 계수가 1인경우에는 생략가능하다.

올바른 화학 반응식의 계수는 다음과 같은 조건을 만족한다.

  1. 계수는 모두 양의 정수이다.
  2. 계수는 서로소이다. 즉, 계수의 최대공약수는 1이다.
  3. 모든 원소의 좌변의 양과 우변의 양은 같아야 한다.

화학 반응식이 위의 세가지 조건을 만족하는 경우에는 올바른 화학 반응식이라고 한다. 화학에서 실제로 일어날 수 없는 반응이나 존재하지 않는 분자가 등장한다고 해도, 이 문제에서는 올바른 화학 반응식이다. (H2 → H2도 올바른 반응식이다)

조건 1과 3(2는 필수는 아니다)을 만족하는 화학 반응식은 균형 반응식이라고 부른다.

화학 반응식 4와 5같이 계수가 없는 화학 반응식이 주어졌을 때, 계수를 올바르게 구하는 프로그램을 작성하시오.

모든 올바른 화학 반응식의 계수가 유일하게 결정되는 것은 아니다. 예를 들어, H2O와 NH3를 생성하는 반응식이 다음과 같다고 하자.

xH2 + yO2 + zN2 + uH2 → vH2O + wNH3

이 경우에 (x,y,z,u,v,w) = (2,1,1,3,2,2)가 답이 된다. 하지만, (4,2,1,3,4,2)와 (4,2,3,9,4,6)도 올바른 정답이 된다. 하지만, 이 문제에서 주어지는 모든 화학 반응식의 계수는 유일하게 결정된다. 

입력

입력은 여러 개의 계수가 없는 화학 반응식으로 이루어져 있다.

화학 반응식은 아래와 같이 BNF로 나타낼 수 있다.

<chemical_equation> ::= <molecule_sequence> "->" <molecule_sequence>
<molecule_sequence> ::= <molecule> | <molecule> "+" <molecule_sequence>
         <molecule> ::= <group> | <group> <molecule>
            <group> ::= <unit_group> | <unit_group> <number>
       <unit_group> ::= <chemical_element> | "(" <molecule> ")"
 <chemical_element> ::= <uppercase_letter>
                      | <uppercase_letter> <lowercase_letter>
 <uppercase_letter> ::= "A" | "B" | "C" | "D" | "E" | "F" | "G" | "H" | "I"
                      | "J" | "K" | "L" | "M" | "N" | "O" | "P" | "Q" | "R"
                      | "S" | "T" | "U" | "V" | "W" | "X" | "Y" | "Z"
 <lowercase_letter> ::= "a" | "b" | "c" | "d" | "e" | "f" | "g" | "h" | "i"
                      | "j" | "k" | "l" | "m" | "n" | "o" | "p" | "q" | "r"
                      | "s" | "t" | "u" | "v" | "w" | "x" | "y" | "z"
           <number> ::= <non_zero_digit>
                      | <non_zero_digit> <digit>
   <non_zero_digit> ::= "1" | "2" | "3" | "4" | "5" | "6" | "7" | "8" | "9"
            <digit> ::= "0" | <non_zero_digit>            

모든 화학 반응식의 끝에는 .이 주어진다. 예를 들어, 다음과 같은 방정식이 있을 때

Ca(OH)2 + CO2 → CaCO3 + H2O

입력으로는 다음과 같이 주어진다.

Ca(OH)2+CO2->CaCO3+H2O.

모든 화학 반응식의 길이는 80자를 넘지 않으며, <number>는 10보다 작다. 괄호는 중첩해서 사용되지 않는다. 화학 반응식의 각 변은 최대 10개의 분자로 이루어져 있다. 또, 계수는 항상 40,000을 넘지 않는다.

입력의 마지막 줄에는 .이 하나 주어진다.

위의 설명 중 <chemical_element>에서 주어지는 화학 원소는 존재하지 않는 원소나 아직 발견되지 않는 원소가 주어질 수도 있다. 또, 우누녹튬(Uuo, 118번)과 같이 원소가 알파벳 세글자인 경우도 주어지지 않는다.

출력

각각의 화학 방정식에 대해서, 그 방정식을 올바르게 만드는 계수를 공백으로 구분하여 출력한다.

예제 입력 1

N2+H2->NH3.
Na+Cl2->NaCl.
Ca(OH)2+CO2->CaCO3+H2O.
CaCl2+AgNO3->Ca(NO3)2+AgCl.
C2H5OH+O2->CO2+H2O.
C4H10+O2->CO2+H2O.
A12B23+C34D45+ABCD->A6D7+B8C9.
A98B+B98C+C98->A98B99C99.
A2+B3+C5+D7+E11+F13->ABCDEF.
.

예제 출력 1

1 3 2
2 1 2
1 1 1 1
1 2 1 2
1 3 2 3
2 13 8 10
2 123 33042 5511 4136
1 1 1 1
15015 10010 6006 4290 2730 2310 30030
[{"problem_id":"3880","problem_lang":"0","title":"\ud654\ud559\uc790\uc758 \uc218\ud559","description":"<p>\ubcf4\ud1b5 \uace0\ub4f1\ud559\uad50 \ud654\ud559 \uc2dc\uac04\uc5d0\ub294 \ud654\ud559 \ubc18\uc751\uc2dd\uc744 \ubc30\uc6b4\ub2e4. \uc774 \uc2dd\uc740 \ub2e4\uc74c\uacfc \uac19\uc774 \uc0dd\uacbc\ub2e4.<\/p>\r\n\r\n<pre>\r\n(1) 2H<sub>2<\/sub> + O<sub>2<\/sub> &rarr; 2H<sub>2<\/sub>O\r\n(2) Ca(OH)<sub>2<\/sub> + CO<sub>2<\/sub> &rarr; CaCO<sub>3<\/sub> + H<sub>2<\/sub>O\r\n(3) N<sub>2<\/sub> + 3H<sub>2<\/sub> &rarr; 2NH<sub>3<\/sub><\/pre>\r\n\r\n<p>\uc704\uc758 \uc2dd (1)~(3)\uc740 \ubaa8\ub450 \uc88c\ubcc0\uacfc \uc6b0\ubcc0\uc758 \uade0\ud615\uc774 \ub9de\ucd94\uc5b4\uc838 \uc788\ub2e4. &nbsp;\ud558\uc9c0\ub9cc, \uc544\ub798 \ubc18\uc751\uc2dd\uc740 \uadf8\ub807\uc9c0 \uc54a\ub2e4.<\/p>\r\n\r\n<pre>\r\n(4) Al + O<sub>2<\/sub> &rarr; Al<sub>2<\/sub>O<sub>3<\/sub>\r\n(5) C<sub>3<\/sub>H<sub>8<\/sub> + O<sub>2<\/sub> &rarr; CO<sub>2<\/sub> + H<sub>2<\/sub>O<\/pre>\r\n\r\n<p>\ubaa8\ub4e0 \ud654\ud559 \ubc18\uc751\uc2dd\uc740 \uc9c8\ub7c9 \ubcf4\uc874\uc758 \ubc95\uce59\uc744 \uc9c0\ucf1c\uc57c \ud55c\ub2e4. \uc989, \uac01 \uc6d0\uc18c(H, O, Ca, Al, ...)\uc758 \uac1c\uc218\ub294 \ud654\ud559 \ubc18\uc751\uc73c\ub85c \ubcc0\ud558\uc9c0 \uc54a\uc544\uc57c \ud55c\ub2e4. \ub530\ub77c\uc11c, \uc88c\ubcc0\uacfc \uc6b0\ubcc0\uc5d0 \ubd84\uc790\uc758 \uac1c\uc218\ub97c \ub9de\ucd94\uc5b4 \uc8fc\uc5b4\uc544 \ud55c\ub2e4.<\/p>\r\n\r\n<pre>\r\n(6) 4Al + 3O<sub>2<\/sub> &rarr; 2Al<sub>2<\/sub>O<sub>3<\/sub>\r\n(7) C<sub>3<\/sub>H<sub>8<\/sub> + 5O<sub>2<\/sub> &rarr; 3CO<sub>2<\/sub> + 4H<sub>2<\/sub>O<\/pre>\r\n\r\n<p>\ubc18\uc751\uc2dd (6)\uc758 \uacc4\uc218\ub294 \uc67c\ucabd\ubd80\ud130 \uc624\ub978\ucabd\uae4c\uc9c0 (4,3,2)\uc774\uace0, \ubc18\uc751\uc2dd (7)\uc758 \uacc4\uc218\ub294 (1,5,3,4)\uc774\ub2e4. \uacc4\uc218\uac00 1\uc778\uacbd\uc6b0\uc5d0\ub294 \uc0dd\ub7b5\uac00\ub2a5\ud558\ub2e4.<\/p>\r\n\r\n<p>\uc62c\ubc14\ub978 \ud654\ud559 \ubc18\uc751\uc2dd\uc758 \uacc4\uc218\ub294 \ub2e4\uc74c\uacfc \uac19\uc740 \uc870\uac74\uc744 \ub9cc\uc871\ud55c\ub2e4.<\/p>\r\n\r\n<ol>\r\n\t<li>\uacc4\uc218\ub294 \ubaa8\ub450 \uc591\uc758 \uc815\uc218\uc774\ub2e4.<\/li>\r\n\t<li>\uacc4\uc218\ub294 \uc11c\ub85c\uc18c\uc774\ub2e4. \uc989, \uacc4\uc218\uc758 \ucd5c\ub300\uacf5\uc57d\uc218\ub294 1\uc774\ub2e4.<\/li>\r\n\t<li>\ubaa8\ub4e0 \uc6d0\uc18c\uc758 \uc88c\ubcc0\uc758 \uc591\uacfc \uc6b0\ubcc0\uc758 \uc591\uc740 \uac19\uc544\uc57c \ud55c\ub2e4.<\/li>\r\n<\/ol>\r\n\r\n<p>\ud654\ud559 \ubc18\uc751\uc2dd\uc774 \uc704\uc758 \uc138\uac00\uc9c0 \uc870\uac74\uc744 \ub9cc\uc871\ud558\ub294 \uacbd\uc6b0\uc5d0\ub294 \uc62c\ubc14\ub978 \ud654\ud559 \ubc18\uc751\uc2dd\uc774\ub77c\uace0 \ud55c\ub2e4. \ud654\ud559\uc5d0\uc11c \uc2e4\uc81c\ub85c \uc77c\uc5b4\ub0a0 \uc218 \uc5c6\ub294 \ubc18\uc751\uc774\ub098 \uc874\uc7ac\ud558\uc9c0 \uc54a\ub294 \ubd84\uc790\uac00 \ub4f1\uc7a5\ud55c\ub2e4\uace0 \ud574\ub3c4, \uc774 \ubb38\uc81c\uc5d0\uc11c\ub294 \uc62c\ubc14\ub978 \ud654\ud559 \ubc18\uc751\uc2dd\uc774\ub2e4. (H<sub>2<\/sub> &rarr; H<sub>2<\/sub>\ub3c4 \uc62c\ubc14\ub978 \ubc18\uc751\uc2dd\uc774\ub2e4)<\/p>\r\n\r\n<p>\uc870\uac74 1\uacfc 3(2\ub294 \ud544\uc218\ub294 \uc544\ub2c8\ub2e4)\uc744 \ub9cc\uc871\ud558\ub294 \ud654\ud559 \ubc18\uc751\uc2dd\uc740 \uade0\ud615 \ubc18\uc751\uc2dd\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\r\n\r\n<p>\ud654\ud559 \ubc18\uc751\uc2dd 4\uc640 5\uac19\uc774 \uacc4\uc218\uac00 \uc5c6\ub294 \ud654\ud559 \ubc18\uc751\uc2dd\uc774 \uc8fc\uc5b4\uc84c\uc744 \ub54c, \uacc4\uc218\ub97c \uc62c\ubc14\ub974\uac8c \uad6c\ud558\ub294 \ud504\ub85c\uadf8\ub7a8\uc744 \uc791\uc131\ud558\uc2dc\uc624.<\/p>\r\n\r\n<p>\ubaa8\ub4e0 \uc62c\ubc14\ub978 \ud654\ud559 \ubc18\uc751\uc2dd\uc758 \uacc4\uc218\uac00 \uc720\uc77c\ud558\uac8c \uacb0\uc815\ub418\ub294 \uac83\uc740 \uc544\ub2c8\ub2e4. \uc608\ub97c \ub4e4\uc5b4, H<sub>2<\/sub>O\uc640 NH<sub>3<\/sub>\ub97c \uc0dd\uc131\ud558\ub294 \ubc18\uc751\uc2dd\uc774 \ub2e4\uc74c\uacfc \uac19\ub2e4\uace0 \ud558\uc790.<\/p>\r\n\r\n<pre>\r\nxH2 + yO2 + zN2 + uH2 &rarr; vH<sub>2<\/sub>O + wNH<sub>3<\/sub><\/pre>\r\n\r\n<p>\uc774 \uacbd\uc6b0\uc5d0 (x,y,z,u,v,w) = (2,1,1,3,2,2)\uac00 \ub2f5\uc774 \ub41c\ub2e4. \ud558\uc9c0\ub9cc, (4,2,1,3,4,2)\uc640 (4,2,3,9,4,6)\ub3c4 \uc62c\ubc14\ub978 \uc815\ub2f5\uc774 \ub41c\ub2e4. \ud558\uc9c0\ub9cc, \uc774 \ubb38\uc81c\uc5d0\uc11c \uc8fc\uc5b4\uc9c0\ub294 \ubaa8\ub4e0 \ud654\ud559 \ubc18\uc751\uc2dd\uc758 \uacc4\uc218\ub294 \uc720\uc77c\ud558\uac8c \uacb0\uc815\ub41c\ub2e4.&nbsp;<\/p>\r\n","input":"<p>\uc785\ub825\uc740 \uc5ec\ub7ec \uac1c\uc758 \uacc4\uc218\uac00 \uc5c6\ub294 \ud654\ud559 \ubc18\uc751\uc2dd\uc73c\ub85c \uc774\ub8e8\uc5b4\uc838 \uc788\ub2e4.<\/p>\r\n\r\n<p>\ud654\ud559 \ubc18\uc751\uc2dd\uc740 \uc544\ub798\uc640 \uac19\uc774 BNF\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4.<\/p>\r\n\r\n<pre>\r\n&lt;chemical_equation&gt; ::= &lt;molecule_sequence&gt; &quot;-&gt;&quot; &lt;molecule_sequence&gt;\r\n&lt;molecule_sequence&gt; ::= &lt;molecule&gt; | &lt;molecule&gt; &quot;+&quot; &lt;molecule_sequence&gt;\r\n         &lt;molecule&gt; ::= &lt;group&gt; | &lt;group&gt; &lt;molecule&gt;\r\n            &lt;group&gt; ::= &lt;unit_group&gt; | &lt;unit_group&gt; &lt;number&gt;\r\n       &lt;unit_group&gt; ::= &lt;chemical_element&gt; | &quot;(&quot; &lt;molecule&gt; &quot;)&quot;\r\n &lt;chemical_element&gt; ::= &lt;uppercase_letter&gt;\r\n                      | &lt;uppercase_letter&gt; &lt;lowercase_letter&gt;\r\n &lt;uppercase_letter&gt; ::= &quot;A&quot; | &quot;B&quot; | &quot;C&quot; | &quot;D&quot; | &quot;E&quot; | &quot;F&quot; | &quot;G&quot; | &quot;H&quot; | &quot;I&quot;\r\n                      | &quot;J&quot; | &quot;K&quot; | &quot;L&quot; | &quot;M&quot; | &quot;N&quot; | &quot;O&quot; | &quot;P&quot; | &quot;Q&quot; | &quot;R&quot;\r\n                      | &quot;S&quot; | &quot;T&quot; | &quot;U&quot; | &quot;V&quot; | &quot;W&quot; | &quot;X&quot; | &quot;Y&quot; | &quot;Z&quot;\r\n &lt;lowercase_letter&gt; ::= &quot;a&quot; | &quot;b&quot; | &quot;c&quot; | &quot;d&quot; | &quot;e&quot; | &quot;f&quot; | &quot;g&quot; | &quot;h&quot; | &quot;i&quot;\r\n                      | &quot;j&quot; | &quot;k&quot; | &quot;l&quot; | &quot;m&quot; | &quot;n&quot; | &quot;o&quot; | &quot;p&quot; | &quot;q&quot; | &quot;r&quot;\r\n                      | &quot;s&quot; | &quot;t&quot; | &quot;u&quot; | &quot;v&quot; | &quot;w&quot; | &quot;x&quot; | &quot;y&quot; | &quot;z&quot;\r\n           &lt;number&gt; ::= &lt;non_zero_digit&gt;\r\n                      | &lt;non_zero_digit&gt; &lt;digit&gt;\r\n   &lt;non_zero_digit&gt; ::= &quot;1&quot; | &quot;2&quot; | &quot;3&quot; | &quot;4&quot; | &quot;5&quot; | &quot;6&quot; | &quot;7&quot; | &quot;8&quot; | &quot;9&quot;\r\n            &lt;digit&gt; ::= &quot;0&quot; | &lt;non_zero_digit&gt;            \r\n<\/pre>\r\n\r\n<p>\ubaa8\ub4e0 \ud654\ud559 \ubc18\uc751\uc2dd\uc758 \ub05d\uc5d0\ub294 .\uc774 \uc8fc\uc5b4\uc9c4\ub2e4. \uc608\ub97c \ub4e4\uc5b4, \ub2e4\uc74c\uacfc \uac19\uc740 \ubc29\uc815\uc2dd\uc774 \uc788\uc744 \ub54c<\/p>\r\n\r\n<p>Ca(OH)<sub>2<\/sub> + CO<sub>2<\/sub> &rarr; CaCO<sub>3<\/sub> + H<sub>2<\/sub>O<\/p>\r\n\r\n<p>\uc785\ub825\uc73c\ub85c\ub294 \ub2e4\uc74c\uacfc \uac19\uc774 \uc8fc\uc5b4\uc9c4\ub2e4.<\/p>\r\n\r\n<pre>\r\nCa(OH)2+CO2-&gt;CaCO3+H2O.<\/pre>\r\n\r\n<p>\ubaa8\ub4e0 \ud654\ud559 \ubc18\uc751\uc2dd\uc758 \uae38\uc774\ub294 80\uc790\ub97c \ub118\uc9c0 \uc54a\uc73c\uba70, &lt;number&gt;\ub294 10\ubcf4\ub2e4 \uc791\ub2e4. \uad04\ud638\ub294 \uc911\ucca9\ud574\uc11c \uc0ac\uc6a9\ub418\uc9c0 \uc54a\ub294\ub2e4. \ud654\ud559 \ubc18\uc751\uc2dd\uc758 \uac01 \ubcc0\uc740 \ucd5c\ub300 10\uac1c\uc758 \ubd84\uc790\ub85c \uc774\ub8e8\uc5b4\uc838 \uc788\ub2e4. \ub610, \uacc4\uc218\ub294 \ud56d\uc0c1 40,000\uc744 \ub118\uc9c0 \uc54a\ub294\ub2e4.<\/p>\r\n\r\n<p>\uc785\ub825\uc758 \ub9c8\uc9c0\ub9c9 \uc904\uc5d0\ub294 .\uc774 \ud558\ub098 \uc8fc\uc5b4\uc9c4\ub2e4.<\/p>\r\n\r\n<p>\uc704\uc758 \uc124\uba85 \uc911 &lt;chemical_element&gt;\uc5d0\uc11c \uc8fc\uc5b4\uc9c0\ub294 \ud654\ud559 \uc6d0\uc18c\ub294 \uc874\uc7ac\ud558\uc9c0 \uc54a\ub294 \uc6d0\uc18c\ub098 \uc544\uc9c1 \ubc1c\uacac\ub418\uc9c0 \uc54a\ub294 \uc6d0\uc18c\uac00 \uc8fc\uc5b4\uc9c8 \uc218\ub3c4 \uc788\ub2e4. \ub610, \uc6b0\ub204\ub179\ud2ac(Uuo, 118\ubc88)\uacfc \uac19\uc774 \uc6d0\uc18c\uac00 \uc54c\ud30c\ubcb3 \uc138\uae00\uc790\uc778 \uacbd\uc6b0\ub3c4 \uc8fc\uc5b4\uc9c0\uc9c0 \uc54a\ub294\ub2e4.<\/p>\r\n","output":"<p>\uac01\uac01\uc758 \ud654\ud559 \ubc29\uc815\uc2dd\uc5d0 \ub300\ud574\uc11c, \uadf8 \ubc29\uc815\uc2dd\uc744 \uc62c\ubc14\ub974\uac8c \ub9cc\ub4dc\ub294 \uacc4\uc218\ub97c \uacf5\ubc31\uc73c\ub85c \uad6c\ubd84\ud558\uc5ec \ucd9c\ub825\ud55c\ub2e4.<\/p>\r\n","hint":"","original":"0","problem_lang_code":"\ud55c\uad6d\uc5b4"},{"problem_id":"3880","problem_lang":"1","title":"Chemist\u2019s Math","description":"<p>You have probably learnt chemical equations (chemical reaction formulae) in your high-school days. The following are some well-known equations.<\/p>\r\n\r\n<pre>\r\n(1) 2H<sub>2<\/sub> + O<sub>2<\/sub> &rarr; 2H<sub>2<\/sub>O\r\n(2) Ca(OH)<sub>2<\/sub> + CO<sub>2<\/sub> &rarr; CaCO<sub>3<\/sub> + H<sub>2<\/sub>O\r\n(3) N<sub>2<\/sub> + 3H<sub>2<\/sub> &rarr; 2NH<sub>3<\/sub><\/pre>\r\n\r\n<p>While Equations (1)&ndash;(3) all have balanced left-hand sides and right-hand sides, the following ones do not.<\/p>\r\n\r\n<pre>\r\n(4) Al + O<sub>2<\/sub> &rarr; Al<sub>2<\/sub>O<sub>3<\/sub>\r\n(5) C<sub>3<\/sub>H<sub>8<\/sub> + O<sub>2<\/sub> &rarr; CO<sub>2<\/sub> + H<sub>2<\/sub>O<\/pre>\r\n\r\n<p>The equations must follow the law of conservation of mass; the quantity of each chemical element (such as H, O, Ca, Al) should not change with chemical reactions. So we should &ldquo;adjust&rdquo; the numbers of molecules on the left-hand side and right-hand side:<\/p>\r\n\r\n<pre>\r\n(6) 4Al + 3O<sub>2<\/sub> &rarr; 2Al<sub>2<\/sub>O<sub>3<\/sub>\r\n(7) C<sub>3<\/sub>H<sub>8<\/sub> + 5O<sub>2<\/sub> &rarr; 3CO<sub>2<\/sub> + 4H<sub>2<\/sub>O<\/pre>\r\n\r\n<p>The coefficients of Equation (6) are (4, 3, 2) from left to right, and those of Equation (7) are (1, 5, 3, 4) from left to right. Note that the coefficient 1 may be omitted from chemical equations.<\/p>\r\n\r\n<p>The coefficients of a correct equation must satisfy the following conditions<\/p>\r\n\r\n<ol>\r\n\t<li>The coefficients are all positive integers.<\/li>\r\n\t<li>The coefficients are relatively prime, that is, their greatest common divisor (g.c.d.) is 1.<\/li>\r\n\t<li>The quantities of each chemical element on the left-hand side and the right-hand side are equal.<\/li>\r\n<\/ol>\r\n\r\n<p>Conversely, if a chemical equation satisfies the above three conditions, we regard it as a correct equation, no matter whether the reaction or its constituent molecules can be chemically realized in the real world, and no matter whether it can be called a reaction (e.g., H2 &rarr; H2 is considered correct). A chemical equation satisfying Conditions 1 and 3 (but not necessarily Condition 2) is called a balanced equation.<\/p>\r\n\r\n<p>Your goal is to read in chemical equations with missing coefficients like Equation (4) and (5), line by line, and output the sequences of coefficients that make the equations correct.<\/p>\r\n\r\n<p>Note that the above three conditions do not guarantee that a correct equation is uniquely determined. For example, if we &ldquo;mix&rdquo; the reactions generating H<sub>2<\/sub>O and NH<sub>3<\/sub>, we would get<\/p>\r\n\r\n<pre>\r\nxH2 + yO2 + zN2 + uH2 &rarr; vH<sub>2<\/sub>O + wNH<sub>3<\/sub><\/pre>\r\n\r\n<p>but (x, y, z, u, v, w) = (2, 1, 1, 3, 2, 2) does not represent a unique correct equation; for instance, (4, 2, 1, 3, 4, 2) and (4, 2, 3, 9, 4, 6) are also &ldquo;correct&rdquo; according to the above definition! However, we guarantee that every chemical equation we give you will lead to a unique correct equation by adjusting their coefficients. In other words, we guarantee that (i) every chemical equation can be balanced with positive coefficients, and that (ii) all balanced equations of the original equation can be obtained by multiplying the coefficients of a unique correct equation by a positive integer.<\/p>\r\n","input":"<p>The input is a sequence of chemical equations (without coefficients) of the following syntax in the Backus-Naur Form:<\/p>\r\n\r\n<pre>\r\n&lt;chemical_equation&gt; ::= &lt;molecule_sequence&gt; &quot;-&gt;&quot; &lt;molecule_sequence&gt;\r\n&lt;molecule_sequence&gt; ::= &lt;molecule&gt; | &lt;molecule&gt; &quot;+&quot; &lt;molecule_sequence&gt;\r\n         &lt;molecule&gt; ::= &lt;group&gt; | &lt;group&gt; &lt;molecule&gt;\r\n            &lt;group&gt; ::= &lt;unit_group&gt; | &lt;unit_group&gt; &lt;number&gt;\r\n       &lt;unit_group&gt; ::= &lt;chemical_element&gt; | &quot;(&quot; &lt;molecule&gt; &quot;)&quot;\r\n &lt;chemical_element&gt; ::= &lt;uppercase_letter&gt;\r\n                      | &lt;uppercase_letter&gt; &lt;lowercase_letter&gt;\r\n &lt;uppercase_letter&gt; ::= &quot;A&quot; | &quot;B&quot; | &quot;C&quot; | &quot;D&quot; | &quot;E&quot; | &quot;F&quot; | &quot;G&quot; | &quot;H&quot; | &quot;I&quot;\r\n                      | &quot;J&quot; | &quot;K&quot; | &quot;L&quot; | &quot;M&quot; | &quot;N&quot; | &quot;O&quot; | &quot;P&quot; | &quot;Q&quot; | &quot;R&quot;\r\n                      | &quot;S&quot; | &quot;T&quot; | &quot;U&quot; | &quot;V&quot; | &quot;W&quot; | &quot;X&quot; | &quot;Y&quot; | &quot;Z&quot;\r\n &lt;lowercase_letter&gt; ::= &quot;a&quot; | &quot;b&quot; | &quot;c&quot; | &quot;d&quot; | &quot;e&quot; | &quot;f&quot; | &quot;g&quot; | &quot;h&quot; | &quot;i&quot;\r\n                      | &quot;j&quot; | &quot;k&quot; | &quot;l&quot; | &quot;m&quot; | &quot;n&quot; | &quot;o&quot; | &quot;p&quot; | &quot;q&quot; | &quot;r&quot;\r\n                      | &quot;s&quot; | &quot;t&quot; | &quot;u&quot; | &quot;v&quot; | &quot;w&quot; | &quot;x&quot; | &quot;y&quot; | &quot;z&quot;\r\n           &lt;number&gt; ::= &lt;non_zero_digit&gt;\r\n                      | &lt;non_zero_digit&gt; &lt;digit&gt;\r\n   &lt;non_zero_digit&gt; ::= &quot;1&quot; | &quot;2&quot; | &quot;3&quot; | &quot;4&quot; | &quot;5&quot; | &quot;6&quot; | &quot;7&quot; | &quot;8&quot; | &quot;9&quot;\r\n            &lt;digit&gt; ::= &quot;0&quot; | &lt;non_zero_digit&gt;            \r\n<\/pre>\r\n\r\n<p>Each chemical equation is followed by a period and a newline. No other characters such as spaces do not appear in the input. For instance, the equation<\/p>\r\n\r\n<p>Ca(OH)<sub>2<\/sub> + CO<sub>2<\/sub> &rarr; CaCO<sub>3<\/sub> + H<sub>2<\/sub>O<\/p>\r\n\r\n<p>is represented as<\/p>\r\n\r\n<pre>\r\nCa(OH)2+CO2-&gt;CaCO3+H2O.<\/pre>\r\n\r\n<p>Each chemical equation is no more than 80 characters long, and as the above syntax implies, the &lt;number&gt;&rsquo;s are less than 100. Parentheses may be used but will not be nested (maybe a good news to some of you!). Each side of a chemical equation consists of no more than 10 top-level molecules. The coefficients that make the equations correct will not exceed 40000. The chemical equations in the input have been chosen so that 32-bit integer arithmetic would suffice with appropriate precautions against possible arithmetic over\ufb02ow. You are free to use 64-bit arithmetic, however.<\/p>\r\n\r\n<p>The end of the input is indicated by a line consisting of a single period.<\/p>\r\n\r\n<p>Note that our definition of &lt;chemical_element&gt; above allows chemical elements that do not exist or unknown as of now, and excludes known chemical elements with three-letter names (e.g.,ununbium (Uub), with the atomic number 112).<\/p>\r\n","output":"<p>For each chemical equation, output a line containing the sequence of positive integer coefficients that make the chemical equation correct. Numbers in a line must be separated by a single space. No extra characters should appear in the output.<\/p>\r\n","hint":"<p>For each chemical equation, output a line containing the sequence of positive integer coefficients that make the chemical equation correct. Numbers in a line must be separated by a single space. No extra characters should appear in the output.<\/p>\r\n","original":"1","problem_lang_code":"\uc601\uc5b4"}]

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ACM-ICPC > Regionals > Asia > Japan > Asia Regional Contest 2009 in Tokyo F번