National Olympiad in Informatics, China, 2007

Little Y has recently been working at a currency exchange center. The currency exchange center only offers two types of vouchers to be exchanged: commemorative voucher A (henceforth known as voucher A) and commemorative voucher B (henceforth known as voucher B). All voucher-holding customers possess their very own account. The quantity of vouchers a customer has may be a real number.

Rising and falling daily with the waves of the market, the two types of vouchers each has their own values at any given time, and each unit of a voucher can be traded that day for some amount of cash. We note that on day $K$, the values of voucher A and voucher B are respectively $A_K$ and $B_K$ (dollars/unit voucher).

To make it more convenient for customers, the exchange center offers a very easy system to make transactions: the ratio exchange method. There are two different aspects to the ratio exchange method:

1. Selling vouchers: the customer provides a real number $OP$ in the range $[0,100]$ as the selling ratio. This means that $OP\mathrm{\%}$ of voucher A and $OP\mathrm{\%}$ of voucher B are exchanged for cash with the rate at that point in time.
2. Buying vouchers: the customer pays $IP$ dollars, and the exchange center takes this money to exchange it for vouchers. Furthermore, the ratio between the value of voucher A and voucher B on day $K$ just happens to be $Rat{e}_K$.

For example, let's assume for the next 3 three days the following changes in the values of $A_K$, $B_K$, and $Rat{e}_K$:

Time	$A_K$	$B_K$	$Rat{e}_K$
Day 1	$1$	$1$	$1$
Day 2	$1$	$2$	$2$
Day 3	$2$	$2$	$3$

Let's say that on one day, a customer has $100$ dollars but no vouchers of any kind. The customer carries out the following transactions:

Time	Transaction	Cash (Dollars)	Voucher A	Voucher B
Initial	None	$100$	$0$	$0$
Day 1	Buy — $100$ dollars	$0$	$50$	$50$
Day 2	Sell — $50\mathrm{\%}$	$75$	$25$	$25$
Day 2	Buy — $60$ dollars	$15$	$55$	$40$
Day 3	Sell — $100\mathrm{\%}$	$205$	$0$	$0$

Note that there may be multiple transactions on a single day.

Little Y is a very economically-minded worker. After a relatively long time conducting operations and market estimates, he already knows within the future $N$ days what the values of vouchers A and B, as well as the rate, will be. He wishes to calculate, if one starts with $S$ dollars, what is the maximum amount of cash (in dollars) that can be obtained after $N$ days.

Input Specification

The first line contains two positive integers $N$ and $S$, representing the number of days that little Y's can foresee and the starting amount of cash respectively.
Within the next $N$ lines, the $K$-th line contains three real numbers $A_K$, $B_K$, and $Rat{e}_K$.

Output Specification

Output a single real number $MaxProfit$, indicating the maximum amount of cash in dollars that can be obtained after all operations on the $N$-th day has finished, accurate to $3$ decimal places. Your answer will be considered correct if its absolute difference with the correct answer is no larger than $0.001$.

Sample Input

3 100
1 1 1
1 2 2
2 2 3

Sample Output

225.000

Explanation

Time	Transaction	Cash (Dollars)	Voucher A	Voucher B
Initial	None	$100$	$0$	$0$
Day 1	Buy — $100$ dollars	$0$	$50$	$50$
Day 2	Sell — $100\mathrm{\%}$	$150$	$0$	$0$
Day 2	Buy — $150$ dollars	$0$	$75$	$37.5$
Day 3	Sell — $100\mathrm{\%}$	$225$	$0$	$0$

Constraints

The test data ensures that the precision needed for calculations will not exceed ${10}^{-7}$.
For $40\mathrm{\%}$ of the test cases, $N\le 10$;
For $60\mathrm{\%}$ of the test cases, $N\le 1000$;
For $100\mathrm{\%}$ of the test cases, $N\le 100000$;
For $100\mathrm{\%}$ of the test cases:

• $0<A_K\le 10$;
• $0<B_K\le 10$;
• $0<Rat{e}_K\le 100$;
• $MaxProfit\le {10}^9$.

Hints

The input may be very large, please use faster methods of input.
There must be an optimal series of transaction satisfying the conditions:

• each buying transaction uses up all of the cash, and
• each selling transaction sells all of the vouchers.

Problem translated to English by Alex.