Problem Statement

$P$ is playing a certain social game.
This problem is about enhancing and evolving the cards used in the game.

A card of a certain character evolves $N$ times before reaching its final form.
A card evolves when a material is used on it in [Evolution Synthesis].
The material required for one evolution is one card of that character in its unevolved form.

Cards also have latent abilities, and a card in the $k$-th evolution state has $S_{k}$ latent abilities.
Latent abilities can be [Awakened] by [Enhancement Synthesis], using a card in the same evolution state as the material.
The number of newly [Awakened] latent abilities is the total number of latent abilities currently [Awakened] on the card used as the material, plus $1$.
However, if as a result of [Enhancement Synthesis] the number of [Awakened] latent abilities would exceed the maximum number of latent abilities that can be [Awakened] in that form, it becomes that maximum number.

On evolution, the card moves up to the next form with the number of its [Awakened] latent abilities neither increased nor decreased.
Note that even if the card used as the material for evolution has some latent abilities [Awakened],
[Evolution Synthesis] does not newly [Awaken] any latent abilities.

Given the above, and assuming that the only cards obtainable in the game are cards in the unevolved form with no latent abilities [Awakened],
find the minimum number of cards that must be obtained to create a card in the final form,
for each number of [Awakened] latent abilities of the final form.

Input

$N$
$S_{0}\ S_{1}\ ...\ S_{N}$

$N$ is a non-negative integer. $S_{k} (0 \le k \le N)$ are also non-negative integers.
$0 \le N \le 10$
$0 \le S_{0} \le S_{1} \le ... \le S_{N} \le 10$

Output

Output the number of cards required to reach the final state, for each case from no latent abilities awakened to all awakened (that is, $S_{N}+1$ values), in ascending order of the number awakened, separated by spaces.
After the last number, print a newline without a trailing space.

Sample

3
1 3 6 9

4 5 7 8 11 12 14 16 19 20

3
0 0 0 0

4

4
0 0 0 0 10

5 10 15 20 25 30 35 40 45 50 55