문제
하바는 최근 학교에서 수열과 함수에 대해 배웠다. 배운 것을 바로 적용하는 성격의 하바는 아래와 같이 길이 $N$의 수열 $A$를 받아서 길이 $N$의 수열 $D$를 반환하는 함수 $d(A)$를 만들었다.
$D_0=A_0$이다.
0ドル<i<N$인 $i$에 대해 $D_i=A_{i}-A_{i-1}$이다.
하지만 이 함수를 가지고 놀던 하바는 함수를 여러 번 적용할 수 있다는 것을 깨닫고, 아래와 같이 길이 $N$의 수열 $A$를 받아서 길이 $N$의 수열 $B$를 반환하는 함수 $f(A)$를 만들었다.
0ドル\le i<N$인 $i$에 대해 $B_{i}=(d^{i}(A))_i$이다.
$d^i$가 무엇을 의미하는지 모른다면 하단의 노트를 참고하자.
이제 조금 더 재밌는 함수를 가진 하바는 가지고 있던 수열 $A$에 함수 $f$를 $K$번 적용하여 $B=f^{K}(A)$를 만들었다.
수열 $B$와 함수를 적용한 횟수 $K$가 주어질 때 하바가 가지고 있던 수열 $A$를 추정해 보자.
출력
만약 조건을 만족하는 수열 $A$가 존재하지 않는다면 첫째 줄에 -1
만약 조건을 만족하는 수열 $A$가 존재한다면 첫째 줄에 수열 $A$의 각 원소 $A_0,ドル $A_1,ドル $\ldots,ドル $A_{N-1}$을 10ドル^9 + 7$로 나눈 나머지를 공백으로 구분하여 출력한다.
만약 가능한 수열이 여러 가지라면 그중 아무거나 하나를 출력한다.
예제 출력 1
복사
1 4 27 256 3125
$A = [1, 4, 27, 256, 3125]$라고 하자. 그러면 함수를 1ドル$번 적용한 결과는 $f(A) = [1, 3, 20, 186, 2248]$이 되고, 여기에 함수를 1ドル$번 더 적용하면 $B = f(f(A)) = [1, 2, 15, 134, 1613]$이 된다.
노트
어떠한 함수 $f$에 대해 $f^{n}$은 다음과 같이 정의한다.
$f^{0}(X) =X$다.
$f^{n}(X) =f(f^{n-1}(X))$다.
[{"problem_id":"33723","problem_lang":"0","title":"[F] Functional Sequence","description":"<p>\ud558\ubc14\ub294 \ucd5c\uadfc \ud559\uad50\uc5d0\uc11c \uc218\uc5f4\uacfc \ud568\uc218\uc5d0 \ub300\ud574 \ubc30\uc6e0\ub2e4. \ubc30\uc6b4 \uac83\uc744 \ubc14\ub85c \uc801\uc6a9\ud558\ub294 \uc131\uaca9\uc758 \ud558\ubc14\ub294 \uc544\ub798\uc640 \uac19\uc774 \uae38\uc774 $N$\uc758 \uc218\uc5f4 $A$\ub97c \ubc1b\uc544\uc11c \uae38\uc774 $N$\uc758 \uc218\uc5f4 $D$\ub97c \ubc18\ud658\ud558\ub294 \ud568\uc218 $d(A)$\ub97c \ub9cc\ub4e4\uc5c8\ub2e4.<\/p>\r\n\r\n<ul>\r\n\t<li>$D_0=A_0$\uc774\ub2e4.<\/li>\r\n\t<li>$0&lt;i&lt;N$\uc778 $i$\uc5d0 \ub300\ud574 $D_i=A_{i}-A_{i-1}$\uc774\ub2e4.<\/li>\r\n<\/ul>\r\n\r\n<p>\ud558\uc9c0\ub9cc \uc774 \ud568\uc218\ub97c \uac00\uc9c0\uace0 \ub180\ub358 \ud558\ubc14\ub294 \ud568\uc218\ub97c \uc5ec\ub7ec \ubc88 \uc801\uc6a9\ud560 \uc218 \uc788\ub2e4\ub294 \uac83\uc744 \uae68\ub2eb\uace0, \uc544\ub798\uc640 \uac19\uc774 \uae38\uc774 $N$\uc758 \uc218\uc5f4 $A$\ub97c \ubc1b\uc544\uc11c \uae38\uc774 $N$\uc758 \uc218\uc5f4 $B$\ub97c \ubc18\ud658\ud558\ub294 \ud568\uc218 $f(A)$\ub97c \ub9cc\ub4e4\uc5c8\ub2e4.<\/p>\r\n\r\n<ul>\r\n\t<li>$0\\le i&lt;N$\uc778 $i$\uc5d0 \ub300\ud574 $B_{i}=(d^{i}(A))_i$\uc774\ub2e4.<\/li>\r\n<\/ul>\r\n\r\n<p>$d^i$\uac00 \ubb34\uc5c7\uc744 \uc758\ubbf8\ud558\ub294\uc9c0 \ubaa8\ub978\ub2e4\uba74 \ud558\ub2e8\uc758 \ub178\ud2b8\ub97c \ucc38\uace0\ud558\uc790.<\/p>\r\n\r\n<p>\uc774\uc81c \uc870\uae08 \ub354 \uc7ac\ubc0c\ub294 \ud568\uc218\ub97c \uac00\uc9c4 \ud558\ubc14\ub294 \uac00\uc9c0\uace0 \uc788\ub358 \uc218\uc5f4 $A$\uc5d0 \ud568\uc218 $f$\ub97c $K$\ubc88 \uc801\uc6a9\ud558\uc5ec $B=f^{K}(A)$\ub97c \ub9cc\ub4e4\uc5c8\ub2e4.<\/p>\r\n\r\n<p>\uc218\uc5f4 $B$\uc640 \ud568\uc218\ub97c \uc801\uc6a9\ud55c \ud69f\uc218 $K$\uac00 \uc8fc\uc5b4\uc9c8 \ub54c \ud558\ubc14\uac00 \uac00\uc9c0\uace0 \uc788\ub358 \uc218\uc5f4 $A$\ub97c \ucd94\uc815\ud574 \ubcf4\uc790.<\/p>\r\n","input":"<p>\uccab\uc9f8 \uc904\uc5d0\ub294 \ud558\ubc14\uac00 \ub9cc\ub4e0 \uc218\uc5f4 $B$\uc758 \uae38\uc774 $N$\uacfc \ud558\ubc14\uac00 \ud568\uc218\ub97c \uc801\uc6a9\ud55c \ud69f\uc218 $K$\uac00 \uacf5\ubc31\uc73c\ub85c \uad6c\ubd84\ub418\uc5b4 \uc8fc\uc5b4\uc9c4\ub2e4. $(1 \\le N \\le 5\\,000;$ $0 \\le K \\le 10^9)$<\/p>\r\n\r\n<p>\ub458\uc9f8 \uc904\uc5d0\ub294 \uc218\uc5f4 $B$\uc758 \uc6d0\uc18c $B_0$, $B_1$, $\\ldots$, $B_{N-1}$\uc774 \uacf5\ubc31\uc73c\ub85c \uad6c\ubd84\ub418\uc5b4 \uc8fc\uc5b4\uc9c4\ub2e4. $(0 \\le B_{i} \\le 10^9)$<\/p>\r\n","output":"<p>\ub9cc\uc57d \uc870\uac74\uc744 \ub9cc\uc871\ud558\ub294 \uc218\uc5f4 $A$\uac00 \uc874\uc7ac\ud558\uc9c0 \uc54a\ub294\ub2e4\uba74 \uccab\uc9f8 \uc904\uc5d0 <span style=\"color:#e74c3c;\"><code>-1<\/code><\/span>\uc744 \ucd9c\ub825\ud55c\ub2e4.<\/p>\r\n\r\n<p>\ub9cc\uc57d \uc870\uac74\uc744 \ub9cc\uc871\ud558\ub294 \uc218\uc5f4 $A$\uac00 \uc874\uc7ac\ud55c\ub2e4\uba74 \uccab\uc9f8 \uc904\uc5d0 \uc218\uc5f4 $A$\uc758 \uac01 \uc6d0\uc18c $A_0$, $A_1$, $\\ldots$, $A_{N-1}$\uc744 $10^9 + 7$\ub85c \ub098\ub208 \ub098\uba38\uc9c0\ub97c \uacf5\ubc31\uc73c\ub85c \uad6c\ubd84\ud558\uc5ec \ucd9c\ub825\ud55c\ub2e4.<\/p>\r\n\r\n<p>\ub9cc\uc57d \uac00\ub2a5\ud55c \uc218\uc5f4\uc774 \uc5ec\ub7ec \uac00\uc9c0\ub77c\uba74 \uadf8\uc911 \uc544\ubb34\uac70\ub098 \ud558\ub098\ub97c \ucd9c\ub825\ud55c\ub2e4.<\/p>\r\n","hint":"<p>\uc5b4\ub5a0\ud55c \ud568\uc218 $f$\uc5d0 \ub300\ud574 $f^{n}$\uc740 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<\/p>\r\n\r\n<ul>\r\n\t<li>$f^{0}(X) =X$\ub2e4.<\/li>\r\n\t<li>$f^{n}(X) =f(f^{n-1}(X))$\ub2e4.<\/li>\r\n<\/ul>\r\n","original":"1","html_title":"0","problem_lang_tcode":"Korean","sample_explain_1":"<p>$A = [1, 4, 27, 256, 3125]$\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \ud568\uc218\ub97c $1$\ubc88 \uc801\uc6a9\ud55c \uacb0\uacfc\ub294 $f(A) = [1, 3, 20, 186, 2248]$\uc774 \ub418\uace0, \uc5ec\uae30\uc5d0 \ud568\uc218\ub97c $1$\ubc88 \ub354 \uc801\uc6a9\ud558\uba74 $B = f(f(A)) = [1, 2, 15, 134, 1613]$\uc774 \ub41c\ub2e4.<\/p>\r\n"},{"problem_id":"33723","problem_lang":"1","title":"[F] Functional Sequence","description":"<p>HB has recently learned about functions and sequences at school. HB likes to apply new knowledge immediately so he has created the function $d(A)$ which takes a sequence $A$ of length $N$ and returns a sequence $D$ of length $N$. This function&nbsp;is defined as follows.<\/p>\r\n\r\n<ul>\r\n\t<li>$D_0=A_0$.<\/li>\r\n\t<li>For all $i$ such that $0&lt;i&lt;N$, $D_i=A_{i}-A_{i-1}$.<\/li>\r\n<\/ul>\r\n\r\n<p>While playing with his new function, HB has realized that you can apply this function multiple times to create a new function $f(A)$&nbsp;which&nbsp;takes a sequence $A$ of length $N$ and returns a sequence $B$ of length $N$. This function is defined as follows.<\/p>\r\n\r\n<ul>\r\n\t<li>For all $i$ such that $0\\le i&lt;N$, $B_{i}=(d^{i}(A))_i$.<\/li>\r\n<\/ul>\r\n\r\n<p>If you are unfamiliar with&nbsp;what $d^i$ means, refer to the note section below.<\/p>\r\n\r\n<p>Now that HB has a more interesting function, he has decided to make apply $f$ to his initial sequence $A$ not once but $K$ times to make $B=f^{K}(A)$.<\/p>\r\n\r\n<p>Given $B$ and $K$, let&rsquo;s find HB&rsquo;s initial sequence $A$.<\/p>\r\n","input":"<p>The first line of input contains two space-separated integers $N$ and $K$, denoting the length of the sequence $B$ which HB has created, and the number of times the function was applied, respectively. $(1 \\le N \\le 5\\,000;$ $0 \\le K \\le 10^9)$<\/p>\r\n\r\n<p>the second line of input contains $B_0$, $B_1$, $\\ldots$, $B_{N-1}$, denoting the elements of $B$. There is a space between two elements. $(0 \\le B_{i} \\le 10^9)$<\/p>\r\n","output":"<p>If there is no sequence $A$ which suits the conditions given, print <span style=\"color:#e74c3c;\"><code>-1<\/code><\/span>&nbsp;in the first line.<\/p>\r\n\r\n<p>If there exists a valid sequence $A$, print $A_0$, $A_1$, $\\ldots$, $A_{N-1}$, denoting the elements of $A$. Print each element modulo $10^9 + 7$ with a space between two elements.<\/p>\r\n\r\n<p>If there are multiple valid sequences, print any one of them.<\/p>\r\n","hint":"<p>For a function $f$, $f^{n}$ is defined as follows.<\/p>\r\n\r\n<ul>\r\n\t<li>$f^{0}(X) =X$.<\/li>\r\n\t<li>$f^{n}(X) =f(f^{n-1}(X))$.<\/li>\r\n<\/ul>\r\n","original":"0","html_title":"0","problem_lang_tcode":"English","sample_explain_1":"<p>Let&#39;s say $A = [1, 4, 27, 256, 3125]$. Applying the function $1$ time results in $f(A) = [1, 3, 20, 186, 2248]$, and applying the function $1$ more time results in $B = f(f(A)) = [1, 2, 15, 134, 1613]$.<\/p>\r\n"}]