문제
달구는 지난 2023년과 2024년에 열린 UDPC를 성공적으로 개최한 기념으로 윤이와 포닉스에게 보낼 자그마한 선물을 준비했다. 달구가 준비한 선물은 총 $N$개이고, 각 선물은 1ドル$번부터 $N$번까지의 번호가 붙어 있다.
달구는 준비한 모든 선물을 유니 또는 포닉스에게 보내려고 한다. 그러나 달구는 어떤 선물을 윤이에게 보낼지, 포닉스에게 보낼지를 아직 정하지 못했다. 달구는 $M$개의 조건을 세우고, 이 조건을 모두 만족하는 선에서 윤이와 포닉스에게 선물을 보내려고 한다. 달구가 세운 조건은 아래 세 가지 유형 중 하나이다.
- $a$ $b$
U: $a$번 선물과 $b$번 선물은 모두 윤이에게 보내야 한다.
- $a$ $b$
D: $a$번 선물과 $b$번 선물은 윤이와 포닉스에게 각각 하나씩 보내야 한다.
- $a$ $b$
P: $a$번 선물과 $b$번 선물은 모두 포닉스에게 보내야 한다.
달구가 윤이와 포닉스에게 선물을 보낼 수 있는 서로 다른 방법의 수를 10ドル^9 + 7$로 나눈 나머지를 구해보자. 어떤 두 방법이 서로 다르다는 것은 두 방법에서 윤이 또는 포닉스가 받을 선물 번호의 집합이 서로 다름을 의미한다. 만약 모든 조건을 만족하도록 각 선물을 윤이 또는 포닉스에게 보낼 수 없는 경우에는 대신 0을 출력한다.
출력
달구가 윤이와 포닉스에게 선물을 보낼 수 있는 서로 다른 방법의 수를 10ドル^9 + 7$로 나눈 나머지를 출력한다. 만약 모든 조건을 만족하도록 각 선물을 윤이 또는 포닉스에게 보낼 수 없는 경우에는 대신 0을 출력한다.
[{"problem_id":"33695","problem_lang":"0","title":"\uc120\ubb3c \ubcf4\ub0b4\uae30","description":"<p>\ub2ec\uad6c\ub294 \uc9c0\ub09c 2023\ub144\uacfc 2024\ub144\uc5d0 \uc5f4\ub9b0 UDPC\ub97c \uc131\uacf5\uc801\uc73c\ub85c \uac1c\ucd5c\ud55c \uae30\ub150\uc73c\ub85c \uc724\uc774\uc640 \ud3ec\ub2c9\uc2a4\uc5d0\uac8c \ubcf4\ub0bc \uc790\uadf8\ub9c8\ud55c \uc120\ubb3c\uc744 \uc900\ube44\ud588\ub2e4. \ub2ec\uad6c\uac00 \uc900\ube44\ud55c \uc120\ubb3c\uc740 \ucd1d $N$\uac1c\uc774\uace0, \uac01 \uc120\ubb3c\uc740 $1$\ubc88\ubd80\ud130 $N$\ubc88\uae4c\uc9c0\uc758 \ubc88\ud638\uac00 \ubd99\uc5b4 \uc788\ub2e4.<\/p>\r\n\r\n<p>\ub2ec\uad6c\ub294 \uc900\ube44\ud55c \ubaa8\ub4e0 \uc120\ubb3c\uc744 \uc720\ub2c8 \ub610\ub294 \ud3ec\ub2c9\uc2a4\uc5d0\uac8c \ubcf4\ub0b4\ub824\uace0 \ud55c\ub2e4. \uadf8\ub7ec\ub098 \ub2ec\uad6c\ub294 \uc5b4\ub5a4 \uc120\ubb3c\uc744 \uc724\uc774\uc5d0\uac8c \ubcf4\ub0bc\uc9c0, \ud3ec\ub2c9\uc2a4\uc5d0\uac8c \ubcf4\ub0bc\uc9c0\ub97c \uc544\uc9c1 \uc815\ud558\uc9c0 \ubabb\ud588\ub2e4. \ub2ec\uad6c\ub294 $M$\uac1c\uc758 \uc870\uac74\uc744 \uc138\uc6b0\uace0, \uc774 \uc870\uac74\uc744 \ubaa8\ub450 \ub9cc\uc871\ud558\ub294 \uc120\uc5d0\uc11c \uc724\uc774\uc640 \ud3ec\ub2c9\uc2a4\uc5d0\uac8c \uc120\ubb3c\uc744 \ubcf4\ub0b4\ub824\uace0 \ud55c\ub2e4. \ub2ec\uad6c\uac00 \uc138\uc6b4 \uc870\uac74\uc740 \uc544\ub798 \uc138 \uac00\uc9c0 \uc720\ud615 \uc911 \ud558\ub098\uc774\ub2e4.<\/p>\r\n\r\n<ul>\r\n\t<li>$a$ $b$ <code>U<\/code>: $a$\ubc88 \uc120\ubb3c\uacfc $b$\ubc88 \uc120\ubb3c\uc740 \ubaa8\ub450 <strong>\uc724\uc774<\/strong>\uc5d0\uac8c \ubcf4\ub0b4\uc57c \ud55c\ub2e4.<\/li>\r\n\t<li>$a$ $b$ <code>D<\/code>: $a$\ubc88 \uc120\ubb3c\uacfc $b$\ubc88 \uc120\ubb3c\uc740 \uc724\uc774\uc640 \ud3ec\ub2c9\uc2a4\uc5d0\uac8c <strong>\uac01\uac01 \ud558\ub098\uc529<\/strong> \ubcf4\ub0b4\uc57c \ud55c\ub2e4.<\/li>\r\n\t<li>$a$ $b$ <code>P<\/code>: $a$\ubc88 \uc120\ubb3c\uacfc $b$\ubc88 \uc120\ubb3c\uc740 \ubaa8\ub450 <strong>\ud3ec\ub2c9\uc2a4<\/strong>\uc5d0\uac8c \ubcf4\ub0b4\uc57c \ud55c\ub2e4.<\/li>\r\n<\/ul>\r\n\r\n<p>\ub2ec\uad6c\uac00 \uc724\uc774\uc640 \ud3ec\ub2c9\uc2a4\uc5d0\uac8c \uc120\ubb3c\uc744 \ubcf4\ub0bc \uc218 \uc788\ub294 \uc11c\ub85c \ub2e4\ub978 \ubc29\ubc95\uc758 \uc218\ub97c $10^9 + 7$\ub85c \ub098\ub208 \ub098\uba38\uc9c0\ub97c \uad6c\ud574\ubcf4\uc790. \uc5b4\ub5a4 \ub450 \ubc29\ubc95\uc774 \uc11c\ub85c \ub2e4\ub974\ub2e4\ub294 \uac83\uc740 \ub450 \ubc29\ubc95\uc5d0\uc11c \uc724\uc774 \ub610\ub294 \ud3ec\ub2c9\uc2a4\uac00 \ubc1b\uc744 \uc120\ubb3c \ubc88\ud638\uc758 \uc9d1\ud569\uc774 \uc11c\ub85c \ub2e4\ub984\uc744 \uc758\ubbf8\ud55c\ub2e4. \ub9cc\uc57d \ubaa8\ub4e0 \uc870\uac74\uc744 \ub9cc\uc871\ud558\ub3c4\ub85d \uac01 \uc120\ubb3c\uc744 \uc724\uc774 \ub610\ub294 \ud3ec\ub2c9\uc2a4\uc5d0\uac8c \ubcf4\ub0bc \uc218 \uc5c6\ub294 \uacbd\uc6b0\uc5d0\ub294 \ub300\uc2e0 <span style=\"color:#e74c3c;\"><code>0<\/code><\/span>\uc744 \ucd9c\ub825\ud55c\ub2e4.<\/p>\r\n","input":"<p>\uccab\uc9f8 \uc904\uc5d0 \ub2ec\uad6c\uac00 \uc900\ube44\ud55c \uc120\ubb3c\uc758 \uc218 $N$\uacfc \uc120\ubb3c\uc744 \ubcf4\ub0bc \ub54c \uace0\ub824\ud574\uc57c \ud560 \uc870\uac74\uc758 \uc218 $M$\uc774 \uacf5\ubc31\uc73c\ub85c \uad6c\ubd84\ub418\uc5b4 \uc8fc\uc5b4\uc9c4\ub2e4. $(2 \\le N \\le 200\\ 000; 1 \\le M \\le 200\\ 000)$<\/p>\r\n\r\n<p>\ub2e4\uc74c $M$\uac1c\uc758 \uc904\uc5d0\ub294 \ub450 \uc120\ubb3c\uc758 \ubc88\ud638 $a_i, b_i$\uc640 <code>U<\/code>, <code>D<\/code>, <code>P<\/code> \uc911 \ud558\ub098\uc758 \ubb38\uc790\uac00 \uacf5\ubc31\uc73c\ub85c \uad6c\ubd84\ub418\uc5b4 \uc8fc\uc5b4\uc9c4\ub2e4. $(1 \\le a_i &lt; b_i \\le N)$<\/p>\r\n\r\n<p>\uc785\ub825\uc5d0\uc11c \uc8fc\uc5b4\uc9c0\ub294 \ubaa8\ub4e0 \uc218\ub294 \uc815\uc218\uc774\ub2e4.<\/p>\r\n","output":"<p>\ub2ec\uad6c\uac00 \uc724\uc774\uc640 \ud3ec\ub2c9\uc2a4\uc5d0\uac8c \uc120\ubb3c\uc744 \ubcf4\ub0bc \uc218 \uc788\ub294 \uc11c\ub85c \ub2e4\ub978 \ubc29\ubc95\uc758 \uc218\ub97c $10^9 + 7$\ub85c \ub098\ub208 \ub098\uba38\uc9c0\ub97c \ucd9c\ub825\ud55c\ub2e4. \ub9cc\uc57d \ubaa8\ub4e0 \uc870\uac74\uc744 \ub9cc\uc871\ud558\ub3c4\ub85d \uac01 \uc120\ubb3c\uc744 \uc724\uc774 \ub610\ub294 \ud3ec\ub2c9\uc2a4\uc5d0\uac8c \ubcf4\ub0bc \uc218 \uc5c6\ub294 \uacbd\uc6b0\uc5d0\ub294 \ub300\uc2e0 <span style=\"color:#e74c3c;\"><code>0<\/code><\/span>\uc744 \ucd9c\ub825\ud55c\ub2e4.<\/p>\r\n","hint":"","original":"1","html_title":"0","problem_lang_tcode":"Korean"},{"problem_id":"33695","problem_lang":"1","title":"Gift Distribution","description":"<p>To celebrate the successful hosting of UDPC in both 2023 and 2024, Dalgoo prepared some small gifts for Yunee and Ponix. Dalgoo has prepared a total of $N$ gifts, each labeled with a number from $1$ to $N$.<\/p>\r\n\r\n<p>Dalgoo wants to send each gift&nbsp;to either Yunee or Ponix, but hasn&#39;t yet decided who will receive each gift. To help with the decision, Dalgoo has established $M$ conditions and wants to assign each gift to either Yunee or Ponix in a way that satisfies all of them. Each condition falls into one of the following three types:<\/p>\r\n\r\n<ul>\r\n\t<li>$a$ $b$ <code>U<\/code>: Gifts numbered $a$ and $b$ must both be sent to <strong data-end=\"605\" data-start=\"596\">Yunee<\/strong>.<\/li>\r\n\t<li>$a$ $b$ <code>D<\/code>: Gifts numbered $a$ and $b$ must be sent to Yunee and Ponix, <strong>one by one<\/strong>.<\/li>\r\n\t<li>$a$ $b$ <code>P<\/code>: Gifts numbered $a$ and $b$ must both be sent to <strong data-end=\"756\" data-start=\"747\">Ponix<\/strong>.<\/li>\r\n<\/ul>\r\n\r\n<p>Determine the number of different ways Dalgoo can send gifts to Yunee and Ponix, modulo $10^9 + 7$. Two ways are considered different if the sets of gifts received by Yunee or Ponix are different.&nbsp;If there is no way to send each gift to either Yunee or Ponix while satisfying all conditions, print <span style=\"color:#e74c3c;\"><code>0<\/code><\/span>&nbsp;instead.<\/p>\r\n","input":"<p>The first line of input contains $N$, the number of gifts prepared by Dalgoo, and $M$, the number of conditions to be considered when sending the gifts, separated by a space. $(2 \\le N \\le 200\\ 000; 1 \\le M \\le 200\\ 000)$<\/p>\r\n\r\n<p>Each of the following&nbsp;$M$&nbsp;lines of input contain the numbers $a_i, b_i$ &nbsp;and one of the letters <code>U<\/code>, <code>D<\/code>, <code>P<\/code>, separated by spaces. $(1 \\le a_i&nbsp;&lt; b_i \\le N)$<\/p>\r\n\r\n<p>All numbers given in the input are integers.<\/p>\r\n","output":"<p>Print the number of different ways Dalgoo can send gifts to Yunee and Ponix, modulo $10^9 + 7$.&nbsp;If there is no way to send each gift to either Yunee or Ponix while satisfying all conditions, print <span style=\"color:#e74c3c;\"><code>0<\/code><\/span>&nbsp;instead.<\/p>\r\n","hint":"","original":"0","html_title":"0","problem_lang_tcode":"English"}]