Circuit Connections (Circuit)

Consider an IC (integrated circuit) with $n$ inputs and $n$ outputs, as in the following figure. The $n$ inputs are numbered $1, 2, \ldots, n$ from left to right. Likewise, the $n$ outputs are numbered $1, 2, \ldots, n$ from left to right.

In this IC, each of the $n$ inputs comes out unchanged as one of the $n$ outputs, but you cannot tell which input corresponds to which output. An IC is described by listing, for outputs $1, \ldots, n$, the number of the input that corresponds to each output.

For example, when $n = 3$, there are the following 6 kinds of IC.

Here, connecting 5 copies of IC $(2, 3, 1)$ in series makes the original inputs $3, 1, 2$ correspond to the final outputs $1, 2, 3$ respectively, as in the following figure, so the result behaves the same as IC $(3, 1, 2)$.

Given integers $n$ ($1 \le n \le 10000$), $k$ ($1 \le k \le 10000$), and a sequence of distinct integers $a_1, \ldots, a_n$ from $1$ to $n$, write a program that determines whether connecting $k$ copies of the same kind of IC in series can make the result behave the same as IC $(a_1, \ldots, a_n)$, and if so, outputs an IC that achieves the goal. When multiple ICs satisfy the condition, you may output any of them.

The input consists of $n + 1$ lines.

The first line contains the integers $n$ and $k$, separated by a space.

Line $i + 1$ ($1 \le i \le n$) contains the integer $a_i$.

The program writes its result to standard output.

If connecting $k$ copies of the same kind of IC in series can make the result behave the same as IC $(a_1, \ldots, a_n)$, write an IC satisfying the condition on $n$ lines, from line 1 to line $n$. That is, for $1 \le i \le n$, line $i$ contains the number of the input corresponding to output $i$ of the IC.

If connecting $k$ copies of the same kind of IC in series cannot make the result behave the same as IC $(a_1, \ldots, a_n)$, output only 1 line containing $0$.