F. Wonderful Jump
time limit per test
4 seconds
memory limit per test
128 megabytes
input
standard input
output
standard output

You are given an array of positive integers $$$a_1,a_2,\ldots,a_n$$$ of length $$$n$$$.

In one operation you can jump from index $$$i$$$ to index $$$j$$$ ($$$1 \le i \le j \le n$$$) by paying $$$\min(a_i, a_{i + 1}, \ldots, a_j) \cdot (j - i)^2$$$ eris.

For all $$$k$$$ from $$$1$$$ to $$$n$$$, find the minimum number of eris needed to get from index $$$1$$$ to index $$$k$$$.

Input

The first line contains a single integer $$$n$$$ ($$$2 \le n \le 4 \cdot 10^5$$$).

The second line contains $$$n$$$ integers $$$a_1,a_2,\ldots a_n$$$ ($$$1 \le a_i \le n$$$).

Output

Output $$$n$$$ integers — the $$$k$$$-th integer is the minimum number of eris needed to reach index $$$k$$$ if you start from index $$$1$$$.

Examples
Input
3
2 1 3
Output
0 1 2
Input
6
1 4 1 6 3 2
Output
0 1 2 3 6 8
Input
2
1 2
Output
0 1
Input
4
1 4 4 4
Output
0 1 4 8
Note

In the first example:

  • From $$$1$$$ to $$$1$$$: the cost is $$$0$$$,
  • From $$$1$$$ to $$$2$$$: $$$1 \rightarrow 2$$$ — the cost is $$$\min(2, 1) \cdot (2 - 1) ^ 2=1$$$,
  • From $$$1$$$ to $$$3$$$: $$$1 \rightarrow 2 \rightarrow 3$$$ — the cost is $$$\min(2, 1) \cdot (2 - 1) ^ 2 + \min(1, 3) \cdot (3 - 2) ^ 2 = 1 + 1 = 2$$$.

In the fourth example from $$$1$$$ to $$$4$$$: $$$1 \rightarrow 3 \rightarrow 4$$$ — the cost is $$$\min(1, 4, 4) \cdot (3 - 1) ^ 2 + \min(4, 4) \cdot (4 - 3) ^ 2 = 4 + 4 = 8$$$.