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G - Infinite Knapsack

Score : $600$ points

Problem Statement

There are $N$ kinds of items, each with infinitely many copies. The $i$-th kind of item has a weight of $A_i$, a volume of $B_i$, and a value of $C_i$.

Level $X$ Takahashi can carry items whose total weight is at most $X$ and whose total volume is at most $X$. Under this condition, it is allowed to carry any number of items of the same kind, or omit some kinds of items.

Let $f(X)$ be the maximum total value of items Level $X$ Takahashi can carry. It can be proved that the limit $\displaystyle\lim_{X\to \infty} \frac{f(X)}{X}$ exists. Find this limit.

Constraints

• $1\leq N\leq 2\times 10^5$
• $10^8\leq A_i,B_i,C_i \leq 10^9$
• All values in the input are integers.

Input

The input is given from Standard Input in the following format:

$N$

$A_1$ $B_1$ $C_1$

$A_2$ $B_2$ $C_2$

$\vdots$

$A_N$ $B_N$ $C_N$

Output

Print the answer. Your output will be considered correct if the absolute or relative error from the judge's answer is at most $10^{-6}$.

2
100000000 200000000 100000000
200000000 100000000 100000000

0.6666666666666667

When $X=300000000$, Takahashi can carry items whose total weight is at most $300000000$ and whose total volume is at most $300000000$.

He can carry, for instance, one copy of the $1$-st item and one copy of the $2$-nd item. Then, the total value of the items is $100000000+100000000=200000000$. This is the maximum achievable value, so $\dfrac{f(300000000)}{300000000}=\dfrac{2}{3}$.

It can also be proved that $\displaystyle\lim_{X\to \infty} \frac{f(X)}{X}$ equals $\dfrac{2}{3}$. Thus, the answer is $0.6666666...$.

1
500000000 300000000 123456789

0.2469135780000000