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E. Chain Reaction
time limit per test
3 seconds
memory limit per test
512 megabytes
input
standard input
output
standard output

There are $$$n$$$ monsters standing in a row. The $$$i$$$-th monster has $$$a_i$$$ health points.

Every second, you can choose one alive monster and launch a chain lightning at it. The lightning deals $$$k$$$ damage to it, and also spreads to the left (towards decreasing $$$i$$$) and to the right (towards increasing $$$i$$$) to alive monsters, dealing $$$k$$$ damage to each. When the lightning reaches a dead monster or the beginning/end of the row, it stops. A monster is considered alive if its health points are strictly greater than $$$0$$$.

For example, consider the following scenario: there are three monsters with health equal to $$$[5, 2, 7]$$$, and $$$k = 3$$$. You can kill them all in $$$4$$$ seconds:

  • launch a chain lightning at the $$$3$$$-rd monster, then their health values are $$$[2, -1, 4]$$$;
  • launch a chain lightning at the $$$1$$$-st monster, then their health values are $$$[-1, -1, 4]$$$;
  • launch a chain lightning at the $$$3$$$-rd monster, then their health values are $$$[-1, -1, 1]$$$;
  • launch a chain lightning at the $$$3$$$-th monster, then their health values are $$$[-1, -1, -2]$$$.

For each $$$k$$$ from $$$1$$$ to $$$\max(a_1, a_2, \dots, a_n)$$$, calculate the minimum number of seconds it takes to kill all the monsters.

Input

The first line contains a single integer $$$n$$$ ($$$1 \le n \le 10^5$$$) — the number of monsters.

The second line contains $$$n$$$ integers $$$a_1, a_2, \dots, a_n$$$ ($$$1 \le a_i \le 10^5$$$) — the health points of the $$$i$$$-th monster.

Output

For each $$$k$$$ from $$$1$$$ to $$$\max(a_1, a_2, \dots, a_n)$$$, output the minimum number of seconds it takes to kill all the monsters.

Examples
Input
3
5 2 7
Output
10 6 4 3 2 2 1
Input
4
7 7 7 7
Output
7 4 3 2 2 2 1
Input
10
1 9 7 6 2 4 7 8 1 3
Output
17 9 5 4 3 3 3 2 1