You are an expert competitive programmer. You will be given a problem statement, test case constraints and example test inputs and outputs. Please reason step by step about the solution (that must respect memory and time limits), then provide a complete implementation in python3.

Your solution must read input from standard input (cin), write output to standard output (cout).
Do not include any debug prints or additional output.

Put your final solution within a single code block:
```python
<your code here>
```

Execution time limit: 2.0 seconds
Memory limit: 512.0 MB

# Problem
Monocarp is planning on opening a chemistry lab. During the first month, he's going to distribute solutions of a certain acid.

First, he will sign some contracts with a local chemistry factory. Each contract provides Monocarp with an unlimited supply of some solution of the same acid. The factory provides $$$n$$$ contract options, numbered from $$$1$$$ to $$$n$$$. The $$$i$$$-th solution has a concentration of $$$x_i\%$$$, the contract costs $$$w_i$$$ burles, and Monocarp will be able to sell it for $$$c_i$$$ burles per liter.

Monocarp is expecting $$$k$$$ customers during the first month. Each customer will buy a liter of a $$$y\%$$$-solution, where $$$y$$$ is a real number chosen uniformly at random from $$$0$$$ to $$$100$$$ independently for each customer. More formally, the probability of number $$$y$$$ being less than or equal to some $$$t$$$ is $$$P(y \le t) = \frac{t}{100}$$$.

Monocarp can mix the solution that he signed the contracts with the factory for, at any ratio. More formally, if he has contracts for $$$m$$$ solutions with concentrations $$$x_1, x_2, \dots, x_m$$$, then, for these solutions, he picks their volumes $$$a_1, a_2, \dots, a_m$$$ so that $$$\sum \limits_{i=1}^{m} a_i = 1$$$ (exactly $$$1$$$ since each customer wants exactly one liter of a certain solution).

The concentration of the resulting solution is $$$\sum \limits_{i=1}^{m} x_i \cdot a_i$$$. The price of the resulting solution is $$$\sum \limits_{i=1}^{m} c_i \cdot a_i$$$.

If Monocarp can obtain a solution of concentration $$$y\%$$$, then he will do it while maximizing its price (the cost for the customer). Otherwise, the customer leaves without buying anything, and the price is considered equal to $$$0$$$.

Monocarp wants to sign some contracts with a factory (possibly, none or all of them) so that the expected profit is maximized — the expected total price of the sold solutions for all $$$k$$$ customers minus the total cost of signing the contracts from the factory.

Print the maximum expected profit Monocarp can achieve.

## Input Format
The first line contains two integers $$$n$$$ and $$$k$$$ ($$$1 \le n \le 5000$$$; $$$1 \le k \le 10^5$$$) — the number of contracts the factory provides and the number of customers.

The $$$i$$$-th of the next $$$n$$$ lines contains three integers $$$x_i, w_i$$$ and $$$c_i$$$ ($$$0 \le x_i \le 100$$$; $$$1 \le w_i \le 10^9$$$; $$$1 \le c_i \le 10^5$$$) — the concentration of the solution, the cost of the contract and the cost per liter for the customer, for the $$$i$$$-th contract.

## Output Format
Print a single real number — the maximum expected profit Monocarp can achieve.

Your answer is considered correct if its absolute or relative error does not exceed $$$10^{-6}$$$.

Formally, let your answer be $$$a$$$, and the jury's answer be $$$b$$$. Your answer is accepted if and only if $$$\frac{|a - b|}{\max{(1, |b|)}} \le 10^{-6}$$$.

## Examples
```input
2 10
0 10 20
100 15 20
```
```output
175.000000000000000
```
-----
```input
2 10
0 100 20
100 150 20
```
```output
0.000000000000000
```
-----
```input
6 15
79 5 35
30 13 132
37 3 52
24 2 60
76 18 14
71 17 7
```
```output
680.125000000000000
```
-----
```input
10 15
46 11 11
4 12 170
69 2 130
2 8 72
82 7 117
100 5 154
38 9 146
97 1 132
0 12 82
53 1 144
```
```output
2379.400000000000000
```

Now solve the problem and return the code.