Falling Portals

There are $N$ ($2\le N\le 2\cdot 10^5$) worlds, each with a portal. Initially, world $i$ (for $1 \leq i \leq N$) is at $x$-coordinate $i$, and $y$-coordinate $A_i$ ($1\le A_i\le 10^9$). There is also a cow on each world. At time $0$, all $y$-coordinates are distinct and the worlds start falling: world $i$ moves continuously in the negative-$y$ direction at a speed of $i$ units per second.

At any time when two worlds are at the same $y$-coordinate (possibly a fractional time), the portals align, meaning that a cow on one of the worlds can choose to travel instantaneously to the other world.

For each $i$, the cow on world $i$ wants to travel to world $Q_i$ ($Q_i\neq i$). Determine how long her journey will take if she travels optimally.

Each answer is a fraction $a/b$ where $a$ and $b$ are positive, relatively prime integers, or $-1$ if the journey is impossible.

The first line contains a single integer $N$.

The next line contains $N$ space-separated integers $A_1, A_2, \ldots, A_N$.

The next line contains $N$ space-separated integers $Q_1, Q_2, \ldots, Q_N$.

Print $N$ lines. The $i$-th line contains the journey length for cow $i$.

Consider the answer for the cow originally on world 2. At time $2$ worlds 1 and 2 align, so the cow can travel to world 1. At time $\frac{7}{2}$ worlds 1 and 3 align, so the cow can travel to world 3.