A13200 | Water Balance

There are $n$ water tanks in a row, $i$ -th of them contains $a_i$ liters of water. The tanks are numbered from $1$ to $n$ from left to right.

You can perform the following operation: choose some subsegment $[l, r]$ ( $1\le l \le r \le n$ ), and redistribute water in tanks $l, l+1, \dots, r$ evenly. In other words, replace each of $a_l, a_{l+1}, \dots, a_r$ by $\frac{a_l + a_{l+1} + \dots + a_r}{r-l+1}$ . For example, if for volumes $[1, 3, 6, 7]$ you choose $l = 2, r = 3$ , new volumes of water will be $[1, 4.5, 4.5, 7]$ . You can perform this operation any number of times.

What is the lexicographically smallest sequence of volumes of water that you can achieve?

As a reminder:

A sequence $a$ is lexicographically smaller than a sequence $b$ of the same length if and only if the following holds: in the first (leftmost) position where $a$ and $b$ differ, the sequence $a$ has a smaller element than the corresponding element in $b$ .

The first line contains an integer $n$ ( $1 \le n \le 10^6$ ) — the number of water tanks.

The second line contains $n$ integers $a_1, a_2, \dots, a_n$ ( $1 \le a_i \le 10^6$ ) — initial volumes of water in the water tanks, in liters.

Because of large input, reading input as doubles is not recommended.

Print the lexicographically smallest sequence you can get. In the $i$ -th line print the final volume of water in the $i$ -th tank.

Your answer is considered correct if the absolute or relative error of each $a_i$ does not exceed $10^{-9}$ .

Formally, let your answer be $a_1, a_2, \dots, a_n$ , and the jury's answer be $b_1, b_2, \dots, b_n$ . Your answer is accepted if and only if $\frac{|a_i - b_i|}{\max{(1, |b_i|)}} \le 10^{-9}$ for each $i$ .