Harmony

According to classical harmony theory, when writing a multi-voice arrangement you should avoid so-called parallel fifths. A parallel fifth is the following situation: the distance in semitones between two voices leaves a remainder of $7$ when divided by $12$; then the pitch changes in both voices, and afterwards the distance in semitones between them again leaves a remainder of $7$ when divided by $12$.¹

Parallel fifths occur only when the pitch changes in both voices. That is, if for two consecutive notes the distance between two voices leaves a remainder of $7$ when divided by $12$, but the pitch stays the same in one or both voices, then it is not a parallel fifth.

Write a program that detects parallel fifths in a $K$-voice arrangement.

Pitches are given as integers. The value $0$ denotes the note C of the first octave; a positive integer $k$ denotes a pitch $k$ semitones higher, and a negative integer a pitch that many semitones lower. For example, $4$ corresponds to the note E of the first octave, and $-3$ to the note A of the small octave.²

¹ A remark for readers more interested in music theory: sometimes not only two consecutive perfect fifths are counted as parallel fifths, but also, for example, a perfect fifth followed by a diminished fifth (seven and six semitones respectively). In this problem, only two consecutive perfect fifths count as parallel fifths.

² You do not need to understand the note names to solve this problem.

The first line contains two space-separated integers $N$ and $K$ ($0 \le N \le 100000$, $2 \le K \le 10$) — the number of notes in each voice, $N$, and the number of voices, $K$.

Each of the next $N$ lines contains $K$ space-separated integers $A_{1,i}\ \ldots\ A_{K,i}$, where $A_{j,i}$ is the pitch of the $i$-th note in the $j$-th voice ($-100 \le A_{K,i} \le \ldots \le A_{1,i} \le 100$, for $i \in 1 \ldots N$). The notes on one line sound simultaneously, and notes on consecutive lines sound one immediately after another.

If the arrangement contains no parallel fifths, output the single word POLE.

Otherwise, output one line for each parallel fifth. If a parallel fifth occurs between note rows $i$ and $i+1$ in voices $s$ and $t$ (with $s < t$), output the three space-separated integers $i$, $s$, and $t$.

List all parallel fifths in increasing order of $i$. If two lines have the same $i$, order them by increasing $s$; if two lines have the same $i$ and $s$, order them by increasing $t$.