Clock Hands

An analog clock has three hands (the second hand, the minute hand and the hour hand) that all rotate smoothly. You can measure the two angles between the second hand and the two other hands.

Write a program that finds the earliest time, at or after a given time, where "no two hands overlap each other" and "the two angles between the second hand and the two other hands are equal" both hold.

Figure D.1. Angles between the second hand and the two other hands

Clocks are not limited to 12-hour clocks. The hour hand of an $H$-hour clock goes around once in $H$ hours. The minute hand still goes around once every hour, and the second hand goes around once every minute. At 0:0:0 (midnight) all three hands point straight up.

The input consists of multiple datasets. Each dataset is one line with four integers $H$, $h$, $m$ and $s$, separated by a space. $H$ means the clock is an $H$-hour clock, and $h$, $m$, $s$ are the hour, minute and second of the specified time.

You may assume $2 \le H \le 100$, $0 \le h < H$, $0 \le m < 60$ and $0 \le s < 60$.

The end of the input is a line containing four zeros.

Figure D.2. Examples of an $H$-hour clock (a 6-hour clock and a 15-hour clock)

For each dataset, output on one line the time $T$ at which "no two hands overlap each other" and "the two angles between the second hand and the two other hands are equal" hold for the first time at or after the specified time.

For $T$ being $h_o$:$m_o$:$s_o$ ($s_o$ seconds past $m_o$ minutes past $h_o$ o'clock), output four non-negative integers $h_o$, $m_o$, $n$ and $d$, separated by a space, where $n/d$ is the irreducible fraction representing $s_o$. When $s_o$ is an integer, including 0, let $d$ be 1.

The time is expressed as the remainder modulo $H$ hours. In other words, one second after $(H-1)$:59:59 is 0:0:0, not $H$:0:0.