CF1067D.Computer Game

Ivan plays some computer game. There are $n$ quests in the game. Each quest can be upgraded once, this increases the reward for its completion. Each quest has $3$ parameters $a_{i}$, $b_{i}$, $p_{i}$: reward for completing quest before upgrade, reward for completing quest after upgrade ($a_{i} \lt b_{i}$) and probability of successful completing the quest.

Each second Ivan can try to complete one quest and he will succeed with probability $p_{i}$. In case of success Ivan will get the reward and opportunity to upgrade any one quest (not necessary the one he just completed). In case of failure he gets nothing. Quests do not vanish after completing.

Ivan has $t$ seconds. He wants to maximize expected value of his total gain after $t$ seconds. Help him to calculate this value.

伊万在玩一款电脑游戏。游戏中共有 $n$ 个任务。每个任务最多可升级一次，升级后完成该任务所获得的奖励会增加。每个任务有三个参数：$a_{i}$、$b_{i}$ 和 $p_{i}$，分别表示升级前完成该任务的奖励、升级后完成该任务的奖励（满足 $a_{i} \lt b_{i}$）以及成功完成该任务的概率。

每一秒，伊万可以尝试完成一个任务，且以概率 $p_{i}$ 成功。若成功，他将立即获得对应奖励，并获得一次升级任意一个任务的机会（不一定是刚刚完成的那个任务）；若失败，则什么也得不到。任务在被完成后不会消失。

伊万总共有 $t$ 秒时间。他希望最大化 $t$ 秒后总收益的期望值。请帮他计算这个最大期望值。

First line contains $2$ integers $n$ ($ 1 \le n \le 10^{5}$) and $t$ ($ 1 \le t \le 10^{10}$) — number of quests and total time.

Following $n$ lines contain description of quests. Each description is $3$ numbers $a_{i}$ $b_{i}$ $p_{i}$ ($1 \le a_{i} \lt b_{i} \le 10^{8}$, $0 \lt p_{i} \lt 1$) — reward for completing quest before upgrade, reward for completing quest after upgrade and probability of successful completing of quest. $a_{i}$ and $b_{i}$ are integers. All probabilities are given with at most $9$ decimal places.

第一行包含两个整数 $n$（$1 \le n \le 10^{5}$）和 $t$（$1 \le t \le 10^{10}$）——分别表示任务数量和总时间。

接下来的 $n$ 行描述各个任务。每行包含三个数 $a_{i}$、$b_{i}$、$p_{i}$（其中 $1 \le a_{i} \lt b_{i} \le 10^{8}$，$0 \lt p_{i} \lt 1$）——分别表示在升级前完成该任务所获得的奖励、在升级后完成该任务所获得的奖励，以及成功完成该任务的概率。$a_{i}$ 和 $b_{i}$ 均为整数。所有概率均以最多 9 位小数的形式给出。

Print the expected value.

Your answer will be accepted if 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 considered correct if $\frac{|a-b|}{max⁡(b, \,\, 1)} \le 10^{-6}$.

输出期望值。

若你的答案的绝对误差或相对误差不超过 $10^{-6}$，则视为正确。形式化地，设你的答案为 $a$，评测机的答案为 $b$。当且仅当 $\frac{|a-b|}{\max(b, \,\, 1)} \le 10^{-6}$ 时，你的答案被视为正确。