A group of farmers has some elevation data, and we’re going to help them understand how rainfall flows over their farmland.

We’ll represent the land as a two-dimensional array of altitudes and use the following model, based on the idea that water flows downhill:

If a cell’s four neighboring cells all have higher altitudes, we call this cell a sink; water collects in sinks.

Otherwise, water will flow to the neighboring cell with the lowest altitude. You may assume there is a unique lowest neighbor and that this neighbor will be lower than the cell.

Cells that drain into the same sink – directly or indirectly – are said to be part of the same basin.

Your challenge is to partition the map into basins. In particular, given a map of elevations, your code should partition the map into basins and output the sizes of the basins, in descending order.

Assume the elevation maps are square. Input will begin with a line with one integer, S, the height (and width) of the map. The next S lines will each contain a row of the map, each with S integers – the elevations of the S cells in the row.

Your code should output a space-separated list of the basin sizes, in descending order. (Trailing spaces are ignored.)

A few examples are below.

Input:

3

1 5 2

2 4 7

3 6 9

Output:

7 2

The basins, labeled with A’s and B’s, are:

A A B

A A B

A A A

Input:

1

10

Output:

1

There is only one basin in this case.

Input:

5

1 0 2 5 8

2 3 4 7 9

3 5 7 8 9

1 2 5 4 2

2 3 5 2 1

Output:

11 7 7

The basins, labeled with A’s, B’s, and C’s, are:

A A A A A

A A A A A

B B A C C

B B B C C

B B C C C

Input:

4

0 2 1 3

2 1 0 4

3 3 3 3

5 5 2 1

Output:

7 5 4

The basins, labeled with A’s, B’s, and C’s, are:

A A B B

A B B B

A B B C

A C C C