Index Case

문제

The epidemiologist W. Andy wants to find the index case of an ongoing crisis. To do this, he modelled the city of the outbreak and its $nn$ residents with a cellular automaton. The city is represented by $nn$ cells numbered from $11$ to $nn$ and each cell has two neighbouring cells, one to its left and one to its right. The left neighbour of cell $ii$ is cell $i-1i-1$ and the right neighbour is cell $i+1i+1$. Additionally, the left neighbour of cell $11$ is cell $nn$ and the right neighbour of cell $nn$ is cell $11$. Thus, the city and the corresponding automaton form a simple cycle.

Each cell contains an integer between $11$ and $mm$ which represents how likely it is that this person is infected. Since the virus can only be transmitted by personal contact, the value in the $ii$th cell on day $dd$ only depends on the values of its neighbours and itself on the previous day. If we denote this value by $s_d[i]s\_\{d\}[i]$, then the outbreak can be simulated by a function $ff$ using the formula: \[s_{d}[i]=f\big(s_{d-1}[i-1],s_{d-1}[i],s_{d-1}[i+1]\big).\] Note that as the city is cyclic both $i+1i+1$ and $i-1i-1$ are calculated modulo $nn$.

Andy wants to find the index case, so he first has to find $s_{0}s\_0$, the state of the city on day zero. This poses a problem, however, as it is not known on which day the crisis started. Right now, Andy believes that he accomplished the task and found the state $s_{0}s\_0$, but you are not convinced. Therefore, you want to check if there may be a state previous to the initial state proposed by Andy, i.e. whether there exists any state $s_{-1}s\_\{-1\}$ that gets transformed into $s_{0}s\_0$ by applying $ff$.

입력

The input consists of:

• One line with two integers $nn$ and $mm$ $(3\le n\le 200,2\le m\le 10)(3\backslash leq\; n\; \backslash leq\; 200,\; 2\backslash leq\; m\; \backslash leq\; 10)$, the number of cells and the number of states.
• $m^{3}m^3$ lines describing the values $f(x,y,z)f(x,y,z)$ ($1\le f(x,y,z)\le m1\; \backslash le\; f(x,y,z)\; \backslash le\; m$ for each $1\le x,y,z\le m1\; \backslash le\; x,y,z\; \backslash le\; m$) of the function $ff$ modelling the automaton. The values are given in lexicographic order of the arguments: The first value is $f(1,1,1)f(1,1,1)$, the next is $f(1,1,2)f(1,1,2)$, and so on until $f(1,1,m)f(1,1,m)$, followed by $f(1,2,1)f(1,2,1)$ and so forth. The last value is $f(m,m,m)f(m,m,m)$.
• One line with $nn$ integers $s_0[1],\dots ,s_0[n]s\_0[1],\backslash dots,s\_0[n]$ ($1\le s_0[i]\le m1\; \backslash le\; s\_0[i]\; \backslash le\; m$ for each $ii$), the initial state that has been proposed by Andy.

출력

Output yes if there exists at least one possible previous state and no otherwise.