Pleasant Path

When I walk home, I do not always take the shortest route. I take a route that a) keeps bringing me closer to home and b) is the "pleasantest", meaning the average pleasantness factor of the road segments I pass is as high as possible. Write a program that computes the maximum such average.

The map of my city can be described with $n$ places numbered from $1$ to $n$. Place $1$ is my starting point and place $n$ is my home, and the places are sorted by distance so that a place with a higher number is always closer to home than one with a lower number.

There are also $m$ different "road segments", each going from one place $u\_i$ to another place $v\_i$ and having a pleasantness factor $w\_i$, which might come from unusual trees, a cute cat in a window, or something else pleasant. Since I always want to walk in the direction of home, the description includes only road segments with $u\_i<v\_i$.

Someone with a bit of mathematical interest (if there is such a person in this company) might call this a directed, weighted acyclic graph.

The map in the second sample. The pleasantest path is $1\rightarrow 3\rightarrow 5$.

The first line contains the two integers $n$ and $m$ ($2 \leq n \leq 10^5$ , $1 \leq m \leq 2\cdot 10^5$). Each of the following $m$ lines describes one road segment and contains three integers $u\_i$, $v\_i$, $w\_i$ ($1 \leq u\_i < v\_i \leq n$, $1 \le w\_i \le 2\cdot 10^6$), meaning the road segment goes from place $u\_i$ to place $v\_i$ and has pleasantness factor $w\_i$.

No two road segments connect the same pair of places, and it is guaranteed that place $n$ is reachable from place $1$.

Print one number: the highest achievable average of the pleasantness factors along a path from place 1 to place $n$. The answer is considered correct if it has a relative or absolute error of at most $10^{-6}$.