Limit Cycles of xy (mod x+y)

For any two positive integers x and y define F(x,y) = xy (mod x+y).
Given two initial values x[0] and x[1] we can form a sequence using
the recurrence
                    x[n] = F( x[n-1] , x[n-2] )

For some initial values this produces a trivial outcome.  For example, 
if  x[1] = x[0]  is odd, then  x[n] = x[0]  for all n.  Slightly less
obvious is the fact that if x[0]=2k and x[1]=2k-6 then x[n]=2k-6n for
all n up to k/3, at which point the sequence terminates with a value 
of 0.

Most initial values lead to a fixed point, but some lead to a limit
cycle.  The smallest limit cycle is {5,7,11}.  Of course, it follows
that {7,5,11} is also a limit cycle.  (I wonder if it's true that if
xy = z (mod x+y)   and   xz = y (mod x+z)   then   yz = x (mod y+z).)

Anyway, here are several examples of limit cycles with period 3 (along
with their gcd factorizations):

                 {5,7,11}            1*[5,7,11]
                 {69,99,111}         3*[23,33,37]
                 {87,111,153}        3*[29,37,51]
                 {184,704,776}       8*[23,88,97]
                 {125,475,575}       25*[5,19,23]
                 {384,864,1056}      96*[4,9,11]
                 {315,525,735}       105*[3,5,7]
                 {324,756,864}       108*[3,7,8]
                 {1575,2205,2835}    45*[35,49,63]

The smallest limit cycles with period 4 are

               {96,304,384,464}       16*[6,19,24,29]
               {128,192,256,320}      64*[2,3,4,5]
               {243,1701,1215,2187}   243*[1,7,5,9]
               {495,1375,1815,1045}   55*[9,25,33,19]
               {1331,1815,2783,2541}  121*[11,15,23,21]

There smallest examples with period 5 are

              {124,310,248,434,558}    62*[2,5,4,7,9]
              {243,621,567,549,675}    9*[27,69,63,61,75]
              {392,490,686,980,882}    14*[4,5,7,10,9]       

Here are limit cycles with periods 6, 7, and 10:

              {23,61,59,119,79,95}     1*[23,61,59,119,79,95]

      {77,343,371,161,147,259,315}     7*[11,49,53,23,21,37,45]

 {3087,10143,9261,18963,6615,5733,3087,4851,3969,8379}
                
                              441*[7,23,21,43,15,13,7,11,9,19]

Can anyone supply cycles of lengths 8 or 9?  Also, can anyone tell me 
the longest known limit cycle of this type?  Are there any "primitive"
cycles (i.e., no common factors) other than the two shown above?

See More Results on the Form xy (mod x+y)