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문제

긴 복도에 $N$개의 램프가 일렬로 나열되어 있다. 램프는 왼쪽부터 차례로 1번부터 $N$번까지의 번호가 붙어있다. 각 램프는 off또는 on중 하나의 상태이다.

램프의 상태를 바꾸는 특별한 기작이 있어서, 한 번의 작업으로 다음 셋 중 한 가지 동작을 할 수 있다.

  • $1 \le p \le q \le N$을 만족하는 정수 $p$와 $q$를 골라서 $p$, $p+1$, $\cdots$, $q$를 off 상태로 만든다.
  • $1 \le p \le q \le N$을 만족하는 정수 $p$와 $q$를 골라서 $p$, $p+1$, $\cdots$, $q$를 on 상태로 만든다.
  • $1 \le p \le q \le N$을 만족하는 정수 $p$와 $q$를 골라서 $p$, $p+1$, $\cdots$, $q$의 상태를 바꾼다. (off를 on으로, on을 off로)

처음에 램프의 상태는 길이 $N$의 문자열 $A$로 표현된다. $A$의 $i$ 번째 ($1 \le i \le N$) 문자가 0이면 $i$ 번째 램프가 off 상태인 것이고, 1이면 on 상태인 것이다. 우리는 만들고 싶은 상태가 길이 $N$의 문자열 $B$로 표현 되어 있고, 작업의 수를 최소한으로 하여 만들고 싶다. $B$의 $i$ 번째 ($1 \le i \le N$) 문자가 0이면 $i$ 번째 램프가 off 상태인 것이고, 1이면 on 상태인 것이다.

램프의 수와, 현재 상태와 만들고 싶은 상태가 주어졌을 때, 만들고 싶은 상태로 바꾸는 데에 드는 연산의 수의 최솟값을 출력하여라.

입력

표준 입력에서 다음과 같은 형식으로 주어진다.

$N$

$A$

$B$

출력

표준 출력으로 한 개의 줄을 출력하여라. 이는 원하는 상태를 만들기 위한 연산의 수의 최솟값이다.

제한

  • $1 \le N \le 1\ 000\ 000$.
  • $A$와 $B$는 길이 $N$의 문자열이다.
  • $A$와 $B$를 이루는 문자들은 0 혹은 1이다.

서브태스크

번호배점제한
16

N ≤ 18.

241

N ≤ 2 000.

34

Each character in A is 0.

449

No additional constraints.

예제 입력 1

8
11011100
01101001

예제 출력 1

4

이 입력에서 우리는 원하는 상태를 다음과 같은 방법으로 네 번의 작업으로 만들 수 있다.

  • 1, 2, 3, 4번 램프의 상태를 바꾼다. 램프의 상태는 00101100이 된다.
  • 2번 램프를 on 상태로 만든다. 램프의 상태는 01101100이 된다.
  • 6, 7, 8번 램프의 상태를 바꾼다. 램프의 상태는 01101011이 된다.
  • 6, 7번 램프를 off 상태로 만든다. 램프의 상태는 01101001이 된다.

네 번보다 더 적은 작업으로 원하는 상태를 만들 수 있는 방법은 없으므로, 4를 출력한다.

예제 입력 2

13
1010010010100
0000111001011

예제 출력 2

3

예제 입력 3

18
001100010010000110
110110001000100101

예제 출력 3

5
[{"problem_id":"17675","problem_lang":"0","title":"\ub7a8\ud504","description":"<p>\uae34 \ubcf5\ub3c4\uc5d0 $N$\uac1c\uc758 \ub7a8\ud504\uac00 \uc77c\ub82c\ub85c \ub098\uc5f4\ub418\uc5b4 \uc788\ub2e4. \ub7a8\ud504\ub294 \uc67c\ucabd\ubd80\ud130 \ucc28\ub840\ub85c 1\ubc88\ubd80\ud130 $N$\ubc88\uae4c\uc9c0\uc758 \ubc88\ud638\uac00 \ubd99\uc5b4\uc788\ub2e4. \uac01 \ub7a8\ud504\ub294 off\ub610\ub294 on\uc911 \ud558\ub098\uc758 \uc0c1\ud0dc\uc774\ub2e4.<\/p>\r\n\r\n<p>\ub7a8\ud504\uc758 \uc0c1\ud0dc\ub97c \ubc14\uafb8\ub294 \ud2b9\ubcc4\ud55c \uae30\uc791\uc774 \uc788\uc5b4\uc11c, \ud55c \ubc88\uc758 \uc791\uc5c5\uc73c\ub85c \ub2e4\uc74c \uc14b \uc911 \ud55c \uac00\uc9c0 \ub3d9\uc791\uc744 \ud560 \uc218 \uc788\ub2e4.<\/p>\r\n\r\n<ul>\r\n\t<li>$1 \\le p \\le q \\le N$\uc744 \ub9cc\uc871\ud558\ub294 \uc815\uc218 $p$\uc640 $q$\ub97c \uace8\ub77c\uc11c $p$, $p+1$, $\\cdots$, $q$\ub97c off \uc0c1\ud0dc\ub85c \ub9cc\ub4e0\ub2e4.<\/li>\r\n\t<li>$1 \\le p \\le q \\le N$\uc744 \ub9cc\uc871\ud558\ub294 \uc815\uc218 $p$\uc640 $q$\ub97c \uace8\ub77c\uc11c $p$, $p+1$, $\\cdots$, $q$\ub97c on \uc0c1\ud0dc\ub85c \ub9cc\ub4e0\ub2e4.<\/li>\r\n\t<li>$1 \\le p \\le q \\le N$\uc744 \ub9cc\uc871\ud558\ub294 \uc815\uc218 $p$\uc640 $q$\ub97c \uace8\ub77c\uc11c $p$, $p+1$, $\\cdots$, $q$\uc758 \uc0c1\ud0dc\ub97c \ubc14\uafbc\ub2e4. (off\ub97c on\uc73c\ub85c, on\uc744 off\ub85c)<\/li>\r\n<\/ul>\r\n\r\n<p>\ucc98\uc74c\uc5d0 \ub7a8\ud504\uc758 \uc0c1\ud0dc\ub294 \uae38\uc774 $N$\uc758 \ubb38\uc790\uc5f4 $A$\ub85c \ud45c\ud604\ub41c\ub2e4. $A$\uc758 $i$ \ubc88\uc9f8 ($1 \\le i \\le N$) \ubb38\uc790\uac00 <code>0<\/code>\uc774\uba74 $i$ \ubc88\uc9f8 \ub7a8\ud504\uac00 off \uc0c1\ud0dc\uc778 \uac83\uc774\uace0, <code>1<\/code>\uc774\uba74 on \uc0c1\ud0dc\uc778 \uac83\uc774\ub2e4. \uc6b0\ub9ac\ub294 \ub9cc\ub4e4\uace0 \uc2f6\uc740 \uc0c1\ud0dc\uac00 \uae38\uc774 $N$\uc758 \ubb38\uc790\uc5f4 $B$\ub85c \ud45c\ud604 \ub418\uc5b4 \uc788\uace0, \uc791\uc5c5\uc758 \uc218\ub97c \ucd5c\uc18c\ud55c\uc73c\ub85c \ud558\uc5ec \ub9cc\ub4e4\uace0 \uc2f6\ub2e4. $B$\uc758 $i$ \ubc88\uc9f8 ($1 \\le i \\le N$) \ubb38\uc790\uac00 <code>0<\/code>\uc774\uba74 $i$ \ubc88\uc9f8 \ub7a8\ud504\uac00 off \uc0c1\ud0dc\uc778 \uac83\uc774\uace0, <code>1<\/code>\uc774\uba74 on \uc0c1\ud0dc\uc778 \uac83\uc774\ub2e4.<\/p>\r\n\r\n<p>\ub7a8\ud504\uc758 \uc218\uc640, \ud604\uc7ac \uc0c1\ud0dc\uc640 \ub9cc\ub4e4\uace0 \uc2f6\uc740 \uc0c1\ud0dc\uac00 \uc8fc\uc5b4\uc84c\uc744 \ub54c, \ub9cc\ub4e4\uace0 \uc2f6\uc740 \uc0c1\ud0dc\ub85c \ubc14\uafb8\ub294 \ub370\uc5d0 \ub4dc\ub294 \uc5f0\uc0b0\uc758 \uc218\uc758 \ucd5c\uc19f\uac12\uc744 \ucd9c\ub825\ud558\uc5ec\ub77c.<\/p>\r\n","input":"<p>\ud45c\uc900 \uc785\ub825\uc5d0\uc11c \ub2e4\uc74c\uacfc \uac19\uc740 \ud615\uc2dd\uc73c\ub85c \uc8fc\uc5b4\uc9c4\ub2e4.<\/p>\r\n\r\n<p>$N$<\/p>\r\n\r\n<p>$A$<\/p>\r\n\r\n<p>$B$<\/p>\r\n","output":"<p>\ud45c\uc900 \ucd9c\ub825\uc73c\ub85c \ud55c \uac1c\uc758 \uc904\uc744 \ucd9c\ub825\ud558\uc5ec\ub77c. \uc774\ub294 \uc6d0\ud558\ub294 \uc0c1\ud0dc\ub97c \ub9cc\ub4e4\uae30 \uc704\ud55c \uc5f0\uc0b0\uc758 \uc218\uc758 \ucd5c\uc19f\uac12\uc774\ub2e4.<\/p>\r\n","hint":"","original":"1","html_title":"0","problem_lang_tcode":"Korean","limit":"<ul>\r\n\t<li>$1 \\le N \\le 1\\ 000\\ 000$.<\/li>\r\n\t<li>$A$\uc640 $B$\ub294 \uae38\uc774 $N$\uc758 \ubb38\uc790\uc5f4\uc774\ub2e4.<\/li>\r\n\t<li>$A$\uc640 $B$\ub97c \uc774\ub8e8\ub294 \ubb38\uc790\ub4e4\uc740 <code>0<\/code> \ud639\uc740 <code>1<\/code>\uc774\ub2e4.<\/li>\r\n<\/ul>\r\n","subtask1":"<p>N &le; 18.<\/p>\r\n","subtask2":"<p>N &le; 2 000.<\/p>\r\n","subtask3":"<p>Each character in A is <code>0<\/code>.<\/p>\r\n","subtask4":"<p>No additional constraints.<\/p>\r\n","sample_explain_1":"<p>\uc774 \uc785\ub825\uc5d0\uc11c \uc6b0\ub9ac\ub294 \uc6d0\ud558\ub294 \uc0c1\ud0dc\ub97c \ub2e4\uc74c\uacfc \uac19\uc740 \ubc29\ubc95\uc73c\ub85c \ub124 \ubc88\uc758 \uc791\uc5c5\uc73c\ub85c \ub9cc\ub4e4 \uc218 \uc788\ub2e4.<\/p>\r\n\r\n<ul>\r\n\t<li>1, 2, 3, 4\ubc88 \ub7a8\ud504\uc758 \uc0c1\ud0dc\ub97c \ubc14\uafbc\ub2e4. \ub7a8\ud504\uc758 \uc0c1\ud0dc\ub294 <tt>00101100<\/tt>\uc774 \ub41c\ub2e4.<\/li>\r\n\t<li>2\ubc88 \ub7a8\ud504\ub97c on \uc0c1\ud0dc\ub85c \ub9cc\ub4e0\ub2e4. \ub7a8\ud504\uc758 \uc0c1\ud0dc\ub294 <tt>01101100<\/tt>\uc774 \ub41c\ub2e4.<\/li>\r\n\t<li>6, 7, 8\ubc88 \ub7a8\ud504\uc758 \uc0c1\ud0dc\ub97c \ubc14\uafbc\ub2e4. \ub7a8\ud504\uc758 \uc0c1\ud0dc\ub294 <tt>01101011<\/tt>\uc774 \ub41c\ub2e4.<\/li>\r\n\t<li>6, 7\ubc88 \ub7a8\ud504\ub97c off \uc0c1\ud0dc\ub85c \ub9cc\ub4e0\ub2e4. \ub7a8\ud504\uc758 \uc0c1\ud0dc\ub294 <tt>01101001<\/tt>\uc774 \ub41c\ub2e4.<\/li>\r\n<\/ul>\r\n\r\n<p>\ub124 \ubc88\ubcf4\ub2e4 \ub354 \uc801\uc740 \uc791\uc5c5\uc73c\ub85c \uc6d0\ud558\ub294 \uc0c1\ud0dc\ub97c \ub9cc\ub4e4 \uc218 \uc788\ub294 \ubc29\ubc95\uc740 \uc5c6\uc73c\ubbc0\ub85c, 4\ub97c \ucd9c\ub825\ud55c\ub2e4.<\/p>\r\n"},{"problem_id":"17675","problem_lang":"1","title":"Lamps","description":"<p>There are N lamps in a row at a long hallway. The lamps are numbered from 1 to N. Each lamps has a state of either off or on. There is a special mechanism to change the states of the lamps. In an operation, we can do one of the followings:<\/p>\r\n\r\n<ul>\r\n\t<li>Choose integers p and q with 1 &le; p &le; q &le; N, and make the lamps p, p + 1, . . . , q off.<\/li>\r\n\t<li>Choose integers p and q with 1 &le; p &le; q &le; N, and make the lamps p, p + 1, . . . , q on.<\/li>\r\n\t<li>Choose integers p and q with 1 &le; p &le; q &le; N, and toggle the states of the lamps p, p + 1, . . . , q (off to on, or on to off).<\/li>\r\n<\/ul>\r\n\r\n<p>Current states of the lamps are represented by a string A of length N. The i-th (1 &le; i &le; N) character of A is 0 if the lamp i is off, and 1 if on. We want to make the states of the lamps to be those represented by a string B of length N, with as few operations as possible. The i-th (1 &le; i &le; N) character of B is 0 if we want to make the lamp i off, and 1 if on.<\/p>\r\n\r\n<p>Write a program which, given the number of lamps, the current states and the target states, calculates the minimum number of operations needed to achieve the target states<\/p>\r\n","input":"<p>Read the following data from the standard input.<\/p>\r\n\r\n<pre>\r\nN\r\nA\r\nB<\/pre>\r\n","output":"<p>Write one line to the standard output. The output should contain the minimum number of operations needed to achieve the target states.<\/p>\r\n","hint":"","original":"0","html_title":"0","problem_lang_tcode":"English","limit":"<ul>\r\n\t<li>1 &le; N &le; 1 000 000.<\/li>\r\n\t<li>A and B are strings of length N.<\/li>\r\n\t<li>Each character in A and B is either 0 or 1.<\/li>\r\n<\/ul>\r\n","subtask1":"<p>N &le; 18.<\/p>\r\n","subtask2":"<p>N &le; 2 000.<\/p>\r\n","subtask3":"<p>Each character in A is <code>0<\/code>.<\/p>\r\n","subtask4":"<p>No additional constraints.<\/p>\r\n","sample_explain_1":"<p>In this sample input, we can achieve the target states in 4 operations, for example as follows:<\/p>\r\n\r\n<ol>\r\n\t<li>Toggle the states of the lamps 1, 2, 3 and 4. The states of the lamps become 00101100.<\/li>\r\n\t<li>Make the lamp 2 on. The states of the lamps become 01101100.<\/li>\r\n\t<li>Toggle the states of the lamps 6, 7 and 8. The states of the lamps become 01101011.<\/li>\r\n\t<li>Make the lamps 6 and 7 off. The states of the lamps become 01101001.<\/li>\r\n<\/ol>\r\n\r\n<p>Since it is impossible to achieve the target states in less than 4 operations, output 4<\/p>\r\n"}]

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