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문제

육각형 타일로 이루어진 칸들이 무한하게 놓여 있고, 재현이는 이 중 "시작 칸"이라고 부르는 칸 위에 서 있다. 두 칸은 모서리를 공유하면 이웃하고 있다고 하자. 재현이는 매번 다음 그림과 같이, 1부터 6까지 번호가 붙은 방향 을 따라서 이웃한 칸 중 하나로 이동할 수 있다.

재현이가 총 $N$번 움직였을 때 만드는 칸들로 이루어지는 경로는 영역을 형성한다. $i$번째 움직일 때는 $D[i]$ 방향으로 $L[i]$ 칸을 이동한다. 이 경로는 다음과 같은 특징이 있다.

  • 이 경로는 닫혀있는데, 마지막에 도착하는 칸은 시작 칸과 동일하다는 뜻이다.
  • 이 경로는 단순한데, 맨 처음 시작한 칸만 빼고 모든 칸은 최대 한 번 방문한다는 뜻이다. 맨 처음 시작한 칸은 맨 마지막까지 합쳐서 정확하게 두 번 방문한다.
  • 이 경로는 드러나 있는데, 경로에 포함되는 모든 칸은 최소한 한 개의 경로에 포함되지 않으면서 내부에 있지 않는 칸과 이웃한다.
    • 만약 어떤 칸이 경로에 포함되지 않으면서, 경로에 포함되는 칸을 지나지 않고 방문할 수 있는 칸의 개수가 유한하다면 이 칸은 내부에 있다고 한다.

다음은 재현이가 갈 수 있는 경로 중 하나의 예이다.

  • 1번 칸 (핑크색)이 시작 (그리고 마지막) 칸이다.
  • 옅은 파란색 칸들은 경로에 포함되는 칸들이며, 방문 순서가 칸 안에 쓰여 있다.
  • x표시가 되어 있는 짙은 파란색 칸들은 내부에 있는 칸들이다.

형성된 영역은 경로에 포함되거나, 내부에 있는 모든 칸들로 이루어진다. 영역 안의 칸 $c$의 거리는 영역 안의 칸들 만 방문해서 시작 칸부터 칸 $c$에 도착할 때까지 필요한 움직임의 최소값이다. 영역 안의 칸에 대한 점수는 $A + d \times B$인데, $A$와 $B$ 는 재현이가 미리 정한 상수값이며, $d$는 이 칸의 거리이다. 다음은 위 예제의 경로에 의 해 형성된 영역 안의 각 칸들의 거리를 보여준다.

재현이가 $N$번 움직일 때 만드는 칸들로 이루어진 경로에 의해 형성된 영역의 모든 칸의 점수의 총합을 구하는 프로그램을 작성하시오. 점수의 총합은 매우 큰 값일 수 있으므로, 이 값을 $10^9 + 7$으로 나눈 나머지를 구하시오.

상세 구현

다음 함수를 구현해야 한다.

int draw_territory(int N, int A, int B, int[] D, int[] L)
  • $N$: 움직임의 수.
  • $A$, $B$: 점수 계산에 필요한 상수
  • $D$: 길이 $N$인 배열로, $D[i]$는 $i$번째 움직임의 방향
  • $L$:길이 $N$인 배열로, $L[i]$는 $i$번째 움직임에서 이동한 칸 수
  • 이 함수는 경로에 의해 형성된 영역의 모든 칸의 점수의 총합을 $10^9 + 7$로 나눈 나머지를 리턴한다.
  • 이 함수는 정확하게 한 번 호출된다.

예제

다음 호출을 생각해보자.

draw_territory(17, 2, 3,
               [1, 2, 3, 4, 5, 4, 3, 2, 1, 6, 2, 3, 4, 5, 6, 6, 1],
               [1, 2, 2, 1, 1, 1, 1, 2, 3, 2, 3, 1, 6, 3, 3, 2, 1])

이는 위 예제에서 설명한 것과 동일하다. 다음 표는 영역에서 가능한 거리마다 각 칸의 점수를 보여준다.

거리 칸 수 각 칸의 점수 총 점수
$0$ $1$ $2 + 0 \times 3 = 2$ $1 \times 2 = 2$
$1$ $4$ $2 + 1 \times 3 = 5$ $4 \times 5 = 20$
$2$ $5$ $2 + 2 \times 3 = 8$ $5 \times 8 = 40$
$3$ $6$ $2 + 3 \times 3 = 11$ $6 \times 11 = 66$
$4$ $4$ $2 + 4 \times 3 = 14$ $4 \times 14 = 56$
$5$ $3$ $2 + 5 \times 3 = 17$ $3 \times 17 = 51$
$6$ $4$ $2 + 6 \times 3 = 20$ $4 \times 20 = 80$
$7$ $4$ $2 + 7 \times 3 = 23$ $4 \times 23 = 92$
$8$ $5$ $2 + 8 \times 3 = 26$ $5 \times 26 = 130$
$9$ $3$ $2 + 9 \times 3 = 29$ $3 \times 29 = 87$
$10$ $4$ $2 + 10 \times 3 = 32$ $4 \times 32 = 128$
$11$ $5$ $2 + 11 \times 3 = 35$ $5 \times 35 = 175$
$12$ $2$ $2 + 12 \times 3 = 38$ $2 \times 38 = 76$

점수의 총합은 $2 + 20 + 40 + 66 + 56 + 51 + 80 + 92 + 130 + 87 + 128 + 175 + 76 = 1003$이다. 따라서, draw_territory 함수의 리턴값은 $1003$이어야 한다.

제한

  • $3 \le N ≤ 200 000$
  • $0 \le A,B \le 10$
  • $1 \le D[i] \le 6$ (모든 $0 \le i \le N - 1$)
  • $1 \le L[i]$ (모든 $0 \le i \le N - 1$)
  • $L$의 모든 원소의 합은 $10^9$ 을 넘지 않는다.

서브태스크

번호배점제한
13

$N = 3$, $B = 0$

26

$N = 3$

311

$L$의 모든 원소의 합은 2000을 넘지 않는다.

412

$B = 0$이고 $L$의 모든 원소의 합은 200 000을 넘지 않는다.

515

$B = 0$

619

$L$의 모든 원소의 합은 200 000을 넘지 않는다.

718

$L[i] = L[i + 1]$ (모든 $0 \le i \le N - 2$)

816

추가적인 제약 조건이 없다.

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\uc6c0\uc9c1\uc600\uc744 \ub54c \ub9cc\ub4dc\ub294 \uce78\ub4e4\ub85c \uc774\ub8e8\uc5b4\uc9c0\ub294 \uacbd\ub85c\ub294 \uc601\uc5ed\uc744 \ud615\uc131\ud55c\ub2e4. $i$\ubc88\uc9f8 \uc6c0\uc9c1\uc77c \ub54c\ub294 $D[i]$ \ubc29\ud5a5\uc73c\ub85c $L[i]$ \uce78\uc744 \uc774\ub3d9\ud55c\ub2e4. \uc774 \uacbd\ub85c\ub294 \ub2e4\uc74c\uacfc \uac19\uc740 \ud2b9\uc9d5\uc774 \uc788\ub2e4.<\/p>\r\n\r\n<ul>\r\n\t<li>\uc774 \uacbd\ub85c\ub294 <em>\ub2eb\ud600\uc788\ub294\ub370<\/em>, \ub9c8\uc9c0\ub9c9\uc5d0 \ub3c4\ucc29\ud558\ub294 \uce78\uc740 \uc2dc\uc791 \uce78\uacfc \ub3d9\uc77c\ud558\ub2e4\ub294 \ub73b\uc774\ub2e4.<\/li>\r\n\t<li>\uc774 \uacbd\ub85c\ub294 <em>\ub2e8\uc21c\ud55c\ub370<\/em>, \ub9e8 \ucc98\uc74c \uc2dc\uc791\ud55c \uce78\ub9cc \ube7c\uace0 \ubaa8\ub4e0 \uce78\uc740 \ucd5c\ub300 \ud55c \ubc88 \ubc29\ubb38\ud55c\ub2e4\ub294 \ub73b\uc774\ub2e4. \ub9e8 \ucc98\uc74c \uc2dc\uc791\ud55c \uce78\uc740 \ub9e8 \ub9c8\uc9c0\ub9c9\uae4c\uc9c0 \ud569\uccd0\uc11c \uc815\ud655\ud558\uac8c \ub450 \ubc88 \ubc29\ubb38\ud55c\ub2e4.<\/li>\r\n\t<li>\uc774 \uacbd\ub85c\ub294 <em>\ub4dc\ub7ec\ub098 \uc788\ub294\ub370<\/em>, \uacbd\ub85c\uc5d0 \ud3ec\ud568\ub418\ub294 \ubaa8\ub4e0 \uce78\uc740 \ucd5c\uc18c\ud55c \ud55c \uac1c\uc758 \uacbd\ub85c\uc5d0 \ud3ec\ud568\ub418\uc9c0 \uc54a\uc73c\uba74\uc11c <em>\ub0b4\ubd80<\/em>\uc5d0 \uc788\uc9c0 \uc54a\ub294 \uce78\uacfc \uc774\uc6c3\ud55c\ub2e4.\r\n\t<ul>\r\n\t\t<li>\ub9cc\uc57d \uc5b4\ub5a4 \uce78\uc774 \uacbd\ub85c\uc5d0 \ud3ec\ud568\ub418\uc9c0 \uc54a\uc73c\uba74\uc11c, \uacbd\ub85c\uc5d0 \ud3ec\ud568\ub418\ub294 \uce78\uc744 \uc9c0\ub098\uc9c0 \uc54a\uace0 \ubc29\ubb38\ud560 \uc218 \uc788\ub294 \uce78\uc758 \uac1c\uc218\uac00 \uc720\ud55c\ud558\ub2e4\uba74 \uc774 \uce78\uc740 <em>\ub0b4\ubd80<\/em>\uc5d0 \uc788\ub2e4\uace0 \ud55c\ub2e4.<\/li>\r\n\t<\/ul>\r\n\t<\/li>\r\n<\/ul>\r\n\r\n<p>\ub2e4\uc74c\uc740 \uc7ac\ud604\uc774\uac00 \uac08 \uc218 \uc788\ub294 \uacbd\ub85c \uc911 \ud558\ub098\uc758 \uc608\uc774\ub2e4.<\/p>\r\n\r\n<ul>\r\n\t<li>1\ubc88 \uce78 (\ud551\ud06c\uc0c9)\uc774 \uc2dc\uc791 (\uadf8\ub9ac\uace0 \ub9c8\uc9c0\ub9c9) \uce78\uc774\ub2e4.<\/li>\r\n\t<li>\uc605\uc740 \ud30c\ub780\uc0c9 \uce78\ub4e4\uc740 \uacbd\ub85c\uc5d0 \ud3ec\ud568\ub418\ub294 \uce78\ub4e4\uc774\uba70, \ubc29\ubb38 \uc21c\uc11c\uac00 \uce78 \uc548\uc5d0 \uc4f0\uc5ec \uc788\ub2e4.<\/li>\r\n\t<li>x\ud45c\uc2dc\uac00 \ub418\uc5b4 \uc788\ub294 \uc9d9\uc740 \ud30c\ub780\uc0c9 \uce78\ub4e4\uc740 \ub0b4\ubd80\uc5d0 \uc788\ub294 \uce78\ub4e4\uc774\ub2e4.<\/li>\r\n<\/ul>\r\n\r\n<p style=\"text-align: center;\"><img alt=\"\" src=\"https:\/\/upload.acmicpc.net\/bb67bdab-9db8-46aa-bf4d-71096460317a\/-\/preview\/\" \/><\/p>\r\n\r\n<p>\ud615\uc131\ub41c \uc601\uc5ed\uc740 \uacbd\ub85c\uc5d0 \ud3ec\ud568\ub418\uac70\ub098, \ub0b4\ubd80\uc5d0 \uc788\ub294 \ubaa8\ub4e0 \uce78\ub4e4\ub85c \uc774\ub8e8\uc5b4\uc9c4\ub2e4. \uc601\uc5ed \uc548\uc758 \uce78 $c$\uc758 \uac70\ub9ac\ub294 \uc601\uc5ed \uc548\uc758 \uce78\ub4e4 \ub9cc \ubc29\ubb38\ud574\uc11c \uc2dc\uc791 \uce78\ubd80\ud130 \uce78 $c$\uc5d0 \ub3c4\ucc29\ud560 \ub54c\uae4c\uc9c0 \ud544\uc694\ud55c \uc6c0\uc9c1\uc784\uc758 \ucd5c\uc18c\uac12\uc774\ub2e4. \uc601\uc5ed \uc548\uc758 \uce78\uc5d0 \ub300\ud55c \uc810\uc218\ub294 $A + d \\times B$\uc778\ub370, $A$\uc640 $B$ \ub294 \uc7ac\ud604\uc774\uac00 \ubbf8\ub9ac \uc815\ud55c \uc0c1\uc218\uac12\uc774\uba70, $d$\ub294 \uc774 \uce78\uc758 \uac70\ub9ac\uc774\ub2e4. \ub2e4\uc74c\uc740 \uc704 \uc608\uc81c\uc758 \uacbd\ub85c\uc5d0 \uc758 \ud574 \ud615\uc131\ub41c \uc601\uc5ed \uc548\uc758 \uac01 \uce78\ub4e4\uc758 \uac70\ub9ac\ub97c \ubcf4\uc5ec\uc900\ub2e4.<\/p>\r\n\r\n<p style=\"text-align: center;\"><img alt=\"\" src=\"https:\/\/upload.acmicpc.net\/5b4396df-6e29-4aa4-af4d-49c1866d8959\/-\/preview\/\" \/><\/p>\r\n\r\n<p>\uc7ac\ud604\uc774\uac00 $N$\ubc88 \uc6c0\uc9c1\uc77c \ub54c \ub9cc\ub4dc\ub294 \uce78\ub4e4\ub85c \uc774\ub8e8\uc5b4\uc9c4 \uacbd\ub85c\uc5d0 \uc758\ud574 \ud615\uc131\ub41c \uc601\uc5ed\uc758 \ubaa8\ub4e0 \uce78\uc758 \uc810\uc218\uc758 \ucd1d\ud569\uc744 \uad6c\ud558\ub294 \ud504\ub85c\uadf8\ub7a8\uc744 \uc791\uc131\ud558\uc2dc\uc624. \uc810\uc218\uc758 \ucd1d\ud569\uc740 \ub9e4\uc6b0 \ud070 \uac12\uc77c \uc218 \uc788\uc73c\ubbc0\ub85c, \uc774 \uac12\uc744 $10^9 + 7$\uc73c\ub85c \ub098\ub208 \ub098\uba38\uc9c0\ub97c \uad6c\ud558\uc2dc\uc624.<\/p>\r\n","input":"","output":"","hint":"","original":"1","html_title":"0","problem_lang_tcode":"Korean","limit":"<ul>\r\n\t<li>$3 \\le N &le; 200 000$<\/li>\r\n\t<li>$0 \\le A,B \\le 10$<\/li>\r\n\t<li>$1 \\le D[i] \\le 6$ (\ubaa8\ub4e0 $0 \\le i \\le N - 1$)<\/li>\r\n\t<li>$1 \\le L[i]$ (\ubaa8\ub4e0 $0 \\le i \\le N - 1$)<\/li>\r\n\t<li>$L$\uc758 \ubaa8\ub4e0 \uc6d0\uc18c\uc758 \ud569\uc740 $10^9$ \uc744 \ub118\uc9c0 \uc54a\ub294\ub2e4.<\/li>\r\n<\/ul>\r\n","subtask1":"<p>$N = 3$, $B = 0$<\/p>\r\n","subtask2":"<p>$N = 3$<\/p>\r\n","subtask3":"<p>$L$\uc758 \ubaa8\ub4e0 \uc6d0\uc18c\uc758 \ud569\uc740 2000\uc744 \ub118\uc9c0 \uc54a\ub294\ub2e4.<\/p>\r\n","subtask4":"<p>$B = 0$\uc774\uace0 $L$\uc758 \ubaa8\ub4e0 \uc6d0\uc18c\uc758 \ud569\uc740 200 000\uc744 \ub118\uc9c0 \uc54a\ub294\ub2e4.<\/p>\r\n","subtask5":"<p>$B = 0$<\/p>\r\n","subtask6":"<p>$L$\uc758 \ubaa8\ub4e0 \uc6d0\uc18c\uc758 \ud569\uc740 200 000\uc744 \ub118\uc9c0 \uc54a\ub294\ub2e4.<\/p>\r\n","subtask7":"<p>$L[i] = L[i + 1]$ (\ubaa8\ub4e0 $0 \\le i \\le N - 2$)<\/p>\r\n","subtask8":"<p>\ucd94\uac00\uc801\uc778 \uc81c\uc57d \uc870\uac74\uc774 \uc5c6\ub2e4.<\/p>\r\n","custom_implementation":"<p>\ub2e4\uc74c \ud568\uc218\ub97c \uad6c\ud604\ud574\uc57c \ud55c\ub2e4.<\/p>\r\n\r\n<pre>\r\nint draw_territory(int N, int A, int B, int[] D, int[] L)<\/pre>\r\n\r\n<ul>\r\n\t<li>$N$: \uc6c0\uc9c1\uc784\uc758 \uc218.<\/li>\r\n\t<li>$A$, $B$: \uc810\uc218 \uacc4\uc0b0\uc5d0 \ud544\uc694\ud55c \uc0c1\uc218<\/li>\r\n\t<li>$D$: \uae38\uc774 $N$\uc778 \ubc30\uc5f4\ub85c, $D[i]$\ub294 $i$\ubc88\uc9f8 \uc6c0\uc9c1\uc784\uc758 \ubc29\ud5a5<\/li>\r\n\t<li>$L$:\uae38\uc774 $N$\uc778 \ubc30\uc5f4\ub85c, $L[i]$\ub294 $i$\ubc88\uc9f8 \uc6c0\uc9c1\uc784\uc5d0\uc11c \uc774\ub3d9\ud55c \uce78 \uc218<\/li>\r\n\t<li>\uc774 \ud568\uc218\ub294 \uacbd\ub85c\uc5d0 \uc758\ud574 \ud615\uc131\ub41c \uc601\uc5ed\uc758 \ubaa8\ub4e0 \uce78\uc758 \uc810\uc218\uc758 \ucd1d\ud569\uc744 $10^9 + 7$\ub85c \ub098\ub208 \ub098\uba38\uc9c0\ub97c \ub9ac\ud134\ud55c\ub2e4.<\/li>\r\n\t<li>\uc774 \ud568\uc218\ub294 \uc815\ud655\ud558\uac8c \ud55c \ubc88 \ud638\ucd9c\ub41c\ub2e4.<\/li>\r\n<\/ul>\r\n","custom_sample":"<p>\ub2e4\uc74c \ud638\ucd9c\uc744 \uc0dd\uac01\ud574\ubcf4\uc790.<\/p>\r\n\r\n<pre>\r\ndraw_territory(17, 2, 3,\r\n               [1, 2, 3, 4, 5, 4, 3, 2, 1, 6, 2, 3, 4, 5, 6, 6, 1],\r\n               [1, 2, 2, 1, 1, 1, 1, 2, 3, 2, 3, 1, 6, 3, 3, 2, 1])<\/pre>\r\n\r\n<p>\uc774\ub294 \uc704 \uc608\uc81c\uc5d0\uc11c \uc124\uba85\ud55c \uac83\uacfc \ub3d9\uc77c\ud558\ub2e4. \ub2e4\uc74c \ud45c\ub294 \uc601\uc5ed\uc5d0\uc11c \uac00\ub2a5\ud55c \uac70\ub9ac\ub9c8\ub2e4 \uac01 \uce78\uc758 \uc810\uc218\ub97c \ubcf4\uc5ec\uc900\ub2e4.<\/p>\r\n\r\n<table class=\"table table-bordered table-center-40\">\r\n\t<thead>\r\n\t\t<tr>\r\n\t\t\t<th>\uac70\ub9ac<\/th>\r\n\t\t\t<th>\uce78 \uc218<\/th>\r\n\t\t\t<th>\uac01 \uce78\uc758 \uc810\uc218<\/th>\r\n\t\t\t<th>\ucd1d \uc810\uc218<\/th>\r\n\t\t<\/tr>\r\n\t<\/thead>\r\n\t<tbody>\r\n\t\t<tr>\r\n\t\t\t<td>$0$<\/td>\r\n\t\t\t<td>$1$<\/td>\r\n\t\t\t<td>$2 + 0 \\times 3 = 2$<\/td>\r\n\t\t\t<td>$1 \\times 2 = 2$<\/td>\r\n\t\t<\/tr>\r\n\t\t<tr>\r\n\t\t\t<td>$1$<\/td>\r\n\t\t\t<td>$4$<\/td>\r\n\t\t\t<td>$2 + 1 \\times 3 = 5$<\/td>\r\n\t\t\t<td>$4 \\times 5 = 20$<\/td>\r\n\t\t<\/tr>\r\n\t\t<tr>\r\n\t\t\t<td>$2$<\/td>\r\n\t\t\t<td>$5$<\/td>\r\n\t\t\t<td>$2 + 2 \\times 3 = 8$<\/td>\r\n\t\t\t<td>$5 \\times 8 = 40$<\/td>\r\n\t\t<\/tr>\r\n\t\t<tr>\r\n\t\t\t<td>$3$<\/td>\r\n\t\t\t<td>$6$<\/td>\r\n\t\t\t<td>$2 + 3 \\times 3 = 11$<\/td>\r\n\t\t\t<td>$6 \\times 11 = 66$<\/td>\r\n\t\t<\/tr>\r\n\t\t<tr>\r\n\t\t\t<td>$4$<\/td>\r\n\t\t\t<td>$4$<\/td>\r\n\t\t\t<td>$2 + 4 \\times 3 = 14$<\/td>\r\n\t\t\t<td>$4 \\times 14 = 56$<\/td>\r\n\t\t<\/tr>\r\n\t\t<tr>\r\n\t\t\t<td>$5$<\/td>\r\n\t\t\t<td>$3$<\/td>\r\n\t\t\t<td>$2 + 5 \\times 3 = 17$<\/td>\r\n\t\t\t<td>$3 \\times 17 = 51$<\/td>\r\n\t\t<\/tr>\r\n\t\t<tr>\r\n\t\t\t<td>$6$<\/td>\r\n\t\t\t<td>$4$<\/td>\r\n\t\t\t<td>$2 + 6 \\times 3 = 20$<\/td>\r\n\t\t\t<td>$4 \\times 20 = 80$<\/td>\r\n\t\t<\/tr>\r\n\t\t<tr>\r\n\t\t\t<td>$7$<\/td>\r\n\t\t\t<td>$4$<\/td>\r\n\t\t\t<td>$2 + 7 \\times 3 = 23$<\/td>\r\n\t\t\t<td>$4 \\times 23 = 92$<\/td>\r\n\t\t<\/tr>\r\n\t\t<tr>\r\n\t\t\t<td>$8$<\/td>\r\n\t\t\t<td>$5$<\/td>\r\n\t\t\t<td>$2 + 8 \\times 3 = 26$<\/td>\r\n\t\t\t<td>$5 \\times 26 = 130$<\/td>\r\n\t\t<\/tr>\r\n\t\t<tr>\r\n\t\t\t<td>$9$<\/td>\r\n\t\t\t<td>$3$<\/td>\r\n\t\t\t<td>$2 + 9 \\times 3 = 29$<\/td>\r\n\t\t\t<td>$3 \\times 29 = 87$<\/td>\r\n\t\t<\/tr>\r\n\t\t<tr>\r\n\t\t\t<td>$10$<\/td>\r\n\t\t\t<td>$4$<\/td>\r\n\t\t\t<td>$2 + 10 \\times 3 = 32$<\/td>\r\n\t\t\t<td>$4 \\times 32 = 128$<\/td>\r\n\t\t<\/tr>\r\n\t\t<tr>\r\n\t\t\t<td>$11$<\/td>\r\n\t\t\t<td>$5$<\/td>\r\n\t\t\t<td>$2 + 11 \\times 3 = 35$<\/td>\r\n\t\t\t<td>$5 \\times 35 = 175$<\/td>\r\n\t\t<\/tr>\r\n\t\t<tr>\r\n\t\t\t<td>$12$<\/td>\r\n\t\t\t<td>$2$<\/td>\r\n\t\t\t<td>$2 + 12 \\times 3 = 38$<\/td>\r\n\t\t\t<td>$2 \\times 38 = 76$<\/td>\r\n\t\t<\/tr>\r\n\t<\/tbody>\r\n<\/table>\r\n\r\n<p>\uc810\uc218\uc758 \ucd1d\ud569\uc740 $2 + 20 + 40 + 66 + 56 + 51 + 80 + 92 + 130 + 87 + 128 + 175 + 76 = 1003$\uc774\ub2e4. \ub530\ub77c\uc11c, <code>draw_territory<\/code> \ud568\uc218\uc758 \ub9ac\ud134\uac12\uc740 $1003$\uc774\uc5b4\uc57c \ud55c\ub2e4.<\/p>\r\n","custom_grader":"<p>\uc0d8\ud50c \uadf8\ub808\uc774\ub354\ub294 \ub2e4\uc74c \uc591\uc2dd\uc73c\ub85c \uc785\ub825\uc744 \uc77d\ub294\ub2e4.<\/p>\r\n\r\n<ul>\r\n\t<li>line 1: $N$ $A$ $B$<\/li>\r\n\t<li>line 2 + $i$ ($0 \\le i \\le N - 1$): $D[i]$ $L[i]$<\/li>\r\n<\/ul>\r\n\r\n<p>\uc0d8\ud50c \uadf8\ub808\uc774\ub354\ub294 \ub2e4\uc74c \uc591\uc2dd\uc73c\ub85c \ub2f5\uc744 \ucd9c\ub825\ud55c\ub2e4.<\/p>\r\n\r\n<ul>\r\n\t<li>line 1: <code>draw_territory<\/code>\uc758 \ub9ac\ud134\uac12<\/li>\r\n<\/ul>\r\n","custom_attachment":"<ul>\r\n\t<li><a href=\"https:\/\/upload.acmicpc.net\/50500ff7-ab02-441a-b58a-88bc9bd623f8\/\">hexagon.zip<\/a><\/li>\r\n<\/ul>\r\n"},{"problem_id":"21851","problem_lang":"1","title":"Hexagonal Territory","description":"<p>Pak Dengklek is standing at a cell, called the initial cell, on an infinite hexagonal tiling. Two cells in a hexagonal tiling are said to be neighbouring if they share a common side. In one step, Pak Dengklek can move from one cell to one of its neighbours by moving towards one of the six possible directions, numbered from 1 to 6, as illustrated by the following figure.<\/p>\r\n\r\n<p style=\"text-align: center;\"><img alt=\"\" src=\"https:\/\/upload.acmicpc.net\/763005d4-c8ac-4de4-ac46-4821c6fce16f\/-\/preview\/\" \/><\/p>\r\n\r\n<p>Pak Dengklek will form a territory by following a path that consists of a sequence of cells that are visited by a sequence of $N$ moves. The $i$-th move is made by choosing a direction $D[i]$, then performing $L[i]$ steps in the chosen direction. The path has the following properties:<\/p>\r\n\r\n<ul>\r\n\t<li>The path is&nbsp;<em>closed<\/em>, meaning that the cell at the end of the sequence is the same as the cell at the beginning of the sequence.<\/li>\r\n\t<li>The path is&nbsp;<em>simple<\/em>, meaning that every cell can be visited at most once, except for the initial cell, which is visited exactly twice (at the beginning and at the end).<\/li>\r\n\t<li>The path is&nbsp;<em>exposed<\/em>, meaning that each cell in the path is neighbouring with at least one cell that is not in the path and is not&nbsp;<em>inside<\/em>.\r\n\t<ul>\r\n\t\t<li>A cell is said to be&nbsp;<em>inside<\/em>&nbsp;if it is not in the path and you can only visit a finite number of cells using any sequence of steps without visiting any cell in the path.<\/li>\r\n\t<\/ul>\r\n\t<\/li>\r\n<\/ul>\r\n\r\n<p>The following is an example of a path that can be followed by Pak Dengklek.<\/p>\r\n\r\n<ul>\r\n\t<li>The cell numbered $1$ (shaded pink) is the initial (and final) cell.<\/li>\r\n\t<li>Cells that are numbered (shaded light blue) are cells in the path, numbered in the order they are visited.<\/li>\r\n\t<li>Cells that are crossed (shaded dark blue) are cells that are inside.<\/li>\r\n<\/ul>\r\n\r\n<p style=\"text-align: center;\"><img alt=\"\" src=\"https:\/\/upload.acmicpc.net\/bb67bdab-9db8-46aa-bf4d-71096460317a\/-\/preview\/\" \/><\/p>\r\n\r\n<p>The formed territory will consist of all cells that are in the path or are inside. The distance of a cell $c$ in the territory is the minimum number of steps needed to move from the initial cell to the cell $c$ by visiting only cells in the territory. The score of a cell in the territory is defined as $A + d \\times B$, where $A$ and $B$ are constants predetermined by Pak Dengklek, and $d$ is the distance of the cell in the territory. The following is the illustration of the distance of each cell in the territory formed using the path from the example above.<\/p>\r\n\r\n<p style=\"text-align: center;\"><img alt=\"\" src=\"https:\/\/upload.acmicpc.net\/5b4396df-6e29-4aa4-af4d-49c1866d8959\/-\/preview\/\" \/><\/p>\r\n\r\n<p>Help Pak Dengklek to calculate the total score of all cells in the territory formed by the $N$ moves that he will make. As the total score can be large, calculate it modulo $10^9&nbsp;+ 7$.<\/p>\r\n","input":"","output":"","hint":"","original":"0","html_title":"0","problem_lang_tcode":"English","limit":"<ul>\r\n\t<li>$3 \\le N \\le 200\\,000$<\/li>\r\n\t<li>$0 \\le A, B \\le 10^9$<\/li>\r\n\t<li>$1 \\le D[i] \\le 6$ (for all $0 \\le i \\le N - 1$)<\/li>\r\n\t<li>$1 \\le L[i]$ (for all $0 \\le i \\le N - 1$)<\/li>\r\n\t<li>The sum of all elements of $L$ does not exceed $10^9$.<\/li>\r\n\t<li>The path is closed, simple, and exposed.<\/li>\r\n<\/ul>\r\n","subtask1":"<p>$N = 3$, $B = 0$<\/p>\r\n","subtask2":"<p>$N = 3$<\/p>\r\n","subtask3":"<p>The sum of all elements of $L$ does not exceed $2000$.<\/p>\r\n","subtask4":"<p>$B = 0$, the sum of all elements of $L$ does not exceed $200\\,000$.<\/p>\r\n","subtask5":"<p>$B = 0$<\/p>\r\n","subtask6":"<p>The sum of all elements of $L$ does not exceed $200\\,000$.<\/p>\r\n","subtask7":"<p>$L[i] = L[i + 1]$ (for all $0 \\le i \\le N - 2$)<\/p>\r\n","subtask8":"<p>No additional constraints.<\/p>\r\n","custom_implementation":"<p>You should implement the following procedure.<\/p>\r\n\r\n<pre>\r\n<code>int draw_territory(int N, int A, int B, int[] D, int[] L)\r\n<\/code><\/pre>\r\n\r\n<ul>\r\n\t<li>$N$: the number of moves.<\/li>\r\n\t<li>$A$, $B$: the constants for the calculation of the scores.<\/li>\r\n\t<li>$D$: an array of length $N$, where $D[i]$ is the direction of the $i$-th move.<\/li>\r\n\t<li>$L$: an array of length $N$, where $L[i]$ is the number of steps made by the $i$-th move.<\/li>\r\n\t<li>This procedure should return the total score of the drawn territory modulo $10^9&nbsp;+ 7$.<\/li>\r\n\t<li>This procedure is called exactly once.<\/li>\r\n<\/ul>\r\n","custom_sample":"<p>Consider the following call:<\/p>\r\n\r\n<pre>\r\n<code>draw_territory(17, 2, 3,\r\n               [1, 2, 3, 4, 5, 4, 3, 2, 1, 6, 2, 3, 4, 5, 6, 6, 1],\r\n               [1, 2, 2, 1, 1, 1, 1, 2, 3, 2, 3, 1, 6, 3, 3, 2, 1])\r\n<\/code><\/pre>\r\n\r\n<p>The moves are actually the same as what is illustrated from the description. The following table lists the score of each cell for every possible distance in the territory.<\/p>\r\n\r\n<table class=\"table table-bordered table-center-40\">\r\n\t<thead>\r\n\t\t<tr>\r\n\t\t\t<th>Distance<\/th>\r\n\t\t\t<th>Number of cells<\/th>\r\n\t\t\t<th>Score of each cell<\/th>\r\n\t\t\t<th>Total score<\/th>\r\n\t\t<\/tr>\r\n\t<\/thead>\r\n\t<tbody>\r\n\t\t<tr>\r\n\t\t\t<td>$0$<\/td>\r\n\t\t\t<td>$1$<\/td>\r\n\t\t\t<td>$2 + 0 \\times 3 = 2$<\/td>\r\n\t\t\t<td>$1 \\times 2 = 2$<\/td>\r\n\t\t<\/tr>\r\n\t\t<tr>\r\n\t\t\t<td>$1$<\/td>\r\n\t\t\t<td>$4$<\/td>\r\n\t\t\t<td>$2 + 1 \\times 3 = 5$<\/td>\r\n\t\t\t<td>$4 \\times 5 = 20$<\/td>\r\n\t\t<\/tr>\r\n\t\t<tr>\r\n\t\t\t<td>$2$<\/td>\r\n\t\t\t<td>$5$<\/td>\r\n\t\t\t<td>$2 + 2 \\times 3 = 8$<\/td>\r\n\t\t\t<td>$5 \\times 8 = 40$<\/td>\r\n\t\t<\/tr>\r\n\t\t<tr>\r\n\t\t\t<td>$3$<\/td>\r\n\t\t\t<td>$6$<\/td>\r\n\t\t\t<td>$2 + 3 \\times 3 = 11$<\/td>\r\n\t\t\t<td>$6 \\times 11 = 66$<\/td>\r\n\t\t<\/tr>\r\n\t\t<tr>\r\n\t\t\t<td>$4$<\/td>\r\n\t\t\t<td>$4$<\/td>\r\n\t\t\t<td>$2 + 4 \\times 3 = 14$<\/td>\r\n\t\t\t<td>$4 \\times 14 = 56$<\/td>\r\n\t\t<\/tr>\r\n\t\t<tr>\r\n\t\t\t<td>$5$<\/td>\r\n\t\t\t<td>$3$<\/td>\r\n\t\t\t<td>$2 + 5 \\times 3 = 17$<\/td>\r\n\t\t\t<td>$3 \\times 17 = 51$<\/td>\r\n\t\t<\/tr>\r\n\t\t<tr>\r\n\t\t\t<td>$6$<\/td>\r\n\t\t\t<td>$4$<\/td>\r\n\t\t\t<td>$2 + 6 \\times 3 = 20$<\/td>\r\n\t\t\t<td>$4 \\times 20 = 80$<\/td>\r\n\t\t<\/tr>\r\n\t\t<tr>\r\n\t\t\t<td>$7$<\/td>\r\n\t\t\t<td>$4$<\/td>\r\n\t\t\t<td>$2 + 7 \\times 3 = 23$<\/td>\r\n\t\t\t<td>$4 \\times 23 = 92$<\/td>\r\n\t\t<\/tr>\r\n\t\t<tr>\r\n\t\t\t<td>$8$<\/td>\r\n\t\t\t<td>$5$<\/td>\r\n\t\t\t<td>$2 + 8 \\times 3 = 26$<\/td>\r\n\t\t\t<td>$5 \\times 26 = 130$<\/td>\r\n\t\t<\/tr>\r\n\t\t<tr>\r\n\t\t\t<td>$9$<\/td>\r\n\t\t\t<td>$3$<\/td>\r\n\t\t\t<td>$2 + 9 \\times 3 = 29$<\/td>\r\n\t\t\t<td>$3 \\times 29 = 87$<\/td>\r\n\t\t<\/tr>\r\n\t\t<tr>\r\n\t\t\t<td>$10$<\/td>\r\n\t\t\t<td>$4$<\/td>\r\n\t\t\t<td>$2 + 10 \\times 3 = 32$<\/td>\r\n\t\t\t<td>$4 \\times 32 = 128$<\/td>\r\n\t\t<\/tr>\r\n\t\t<tr>\r\n\t\t\t<td>$11$<\/td>\r\n\t\t\t<td>$5$<\/td>\r\n\t\t\t<td>$2 + 11 \\times 3 = 35$<\/td>\r\n\t\t\t<td>$5 \\times 35 = 175$<\/td>\r\n\t\t<\/tr>\r\n\t\t<tr>\r\n\t\t\t<td>$12$<\/td>\r\n\t\t\t<td>$2$<\/td>\r\n\t\t\t<td>$2 + 12 \\times 3 = 38$<\/td>\r\n\t\t\t<td>$2 \\times 38 = 76$<\/td>\r\n\t\t<\/tr>\r\n\t<\/tbody>\r\n<\/table>\r\n\r\n<p>The total score is $2 + 20 + 40 + 66 + 56 + 51 + 80 + 92 + 130 + 87 + 128 + 175 + 76 = 1003$. Therefore, the procedure&nbsp;<code>draw_territory<\/code>&nbsp;should return $1003$.<\/p>\r\n","custom_grader":"<p>The sample grader reads the input in the following format:<\/p>\r\n\r\n<ul>\r\n\t<li>line $1$: $N \\; A \\; B$<\/li>\r\n\t<li>line $2 + i$ ($0 \\le i \\le N - 1$): $D[i] \\; L[i]$<\/li>\r\n<\/ul>\r\n\r\n<p>The sample grader prints your answer in the following format:<\/p>\r\n\r\n<ul>\r\n\t<li>line $1$: return value of&nbsp;<code>draw_territory<\/code><\/li>\r\n<\/ul>\r\n","custom_attachment":"<ul>\r\n\t<li><a href=\"https:\/\/upload.acmicpc.net\/50500ff7-ab02-441a-b58a-88bc9bd623f8\/\">hexagon.zip<\/a><\/li>\r\n<\/ul>\r\n"}]

샘플 그레이더

샘플 그레이더는 다음 양식으로 입력을 읽는다.

  • line 1: $N$ $A$ $B$
  • line 2 + $i$ ($0 \le i \le N - 1$): $D[i]$ $L[i]$

샘플 그레이더는 다음 양식으로 답을 출력한다.

  • line 1: draw_territory의 리턴값

첨부

제출할 수 있는 언어

C++17, C++14, C++20, C++14 (Clang), C++17 (Clang), C++20 (Clang)

채점 및 기타 정보

  • 예제는 채점하지 않는다.
  • 이 문제의 채점 우선 순위는 2이다.