The recent vote in Puerto Rico favoring United States
statehood has made flag makers very excited. An updated flag
with
$51$ stars rather
than the current one with
$50$ would cause a huge jump in U.S.
flag sales. The current pattern for
$50$ stars is five rows of
$6$ stars, interlaced with
four offset rows of
$5$
stars. The rows alternate until all stars are represented.
* * * * * *
* * * * *
* * * * * *
* * * * *
* * * * * *
* * * * *
* * * * * *
* * * * *
* * * * * *
This pattern has the property that adjacent rows differ by
no more than one star. We represent this star arrangement
compactly by the number of stars in the first two rows:
6,5.
A $51$star flag that
has the same property can have three rows of $9$ stars, interlaced with three rows
of $8$ stars (with a
compact representation of 9,8). Conversely,
if a state were to leave the union, one appealing
representation would be seven rows of seven stars (7,7).
A flag pattern is visually appealing if it
satisfies the following conditions:

Every other row has the same number of stars.

Adjacent rows differ by no more than one star.

The first row cannot have fewer stars than the second
row.
Your team sees beyond the shortterm change to $51$ for the US flag. You want to
corner the market on flags for any union of three or more
states. Given the number $S$ of stars to draw on a flag, find
all possible visually appealing flag patterns.
Input
The input consists of a single line containing the integer
$S$ ($3\le S\le 32\, 767$).
Output
On the first line, print $S$, followed by a colon. Then, for
each visually appealing flag of $S$ stars, print its compact
representation, one per line.
This list of compact representations should be printed in
increasing order of the number of stars in the first row; if
there are ties, print them in order of the number of stars in
the second row. The cases $1$by$S$ and $S$by$1$ are trivial, so do not print those
arrangements.
The compact representations must be printed in the form
“x,y”, with exactly one comma between
x and y and no other
characters.
Sample Input 1 
Sample Output 1 
3

3:
2,1

Sample Input 2 
Sample Output 2 
50

50:
2,1
2,2
3,2
5,4
5,5
6,5
10,10
13,12
17,16
25,25

Sample Input 3 
Sample Output 3 
51

51:
2,1
3,3
9,8
17,17
26,25
