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\%~ of voucher A and ~OP\%~ 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 ~Rate_K~.

For example, let's assume for the next 3 three days the following changes in the values of ~A_K~, ~B_K~, and ~Rate_K~:

Time	~A_K~	~B_K~	~Rate_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\%~	~75~	~25~	~25~
Day 2	Buy — ~60~ dollars	~15~	~55~	~40~
Day 3	Sell — ~100\%~	~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 ~Rate_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\%~	~150~	~0~	~0~
Day 2	Buy — ~150~ dollars	~0~	~75~	~37.5~
Day 3	Sell — ~100\%~	~225~	~0~	~0~

Constraints

The test data ensures that the precision needed for calculations will not exceed ~10^{-7}~.
For ~40\%~ of the test cases, ~N \le 10~;
For ~60\%~ of the test cases, ~N \le 1\,000~;
For ~100\%~ of the test cases, ~N \le 100\,000~;
For ~100\%~ of the test cases:

• ~0 < A_K \le 10~;
• ~0 < B_K \le 10~;
• ~0 < Rate_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.