IBM Ponder This #248 · December 2018

You get an integer number as a challenge and your goal is to make it a square of an integer number.
To do that, you use two circles with four integers each.
You can change the number you got in steps. In each step we multiply it by the number in the bottom of one of two circles and rotate this circle clock-wise to its next number.

For example, if the two circles are [3,14,15,92] and [6,5,3,5] (the first number is initially at the bottom) then we can solve the input number 42 (i.e. make it a square) by multiplying as follows:

1st circle [3,14,15,92]    42*3=126
2nd circle [6,5,3,5]    126*6=756
2nd circle [5,3,5,6]    756*5=3,780
2nd circle [3,5,6,5]    3780*3=11,340
2nd circle [5,6,5,3]    11340*5=56,700
1st circle [14,15,92,3]    56700*14=793,800
2nd circle [6,5,3,5]    793800*6=4,762,800
1st circle [15,92,3,14]    4762800*15=71,442,000
2nd circle [5,3,5,6]    71442000*5=357,210,000
and getting a square (357,210,000 = 18,900**2).

BTW, There is a much simpler solution, can you find it?

Your challenge, this month, is to find two circles (with four numbers each) such that the same initial state will allow you to solve at least 2,187 different integers between 1 and 65,536.
To make the problem more interesting, you must not use "boring" circles. A circle is called boring if the set of solvable numbers using only this circle is closed under multiplication (i.e. if you can solve X and Y using only this circle, then you can also solve X*Y with it). In the example above, one of the circles (which?) is boring.

Solution

Best opened after a real attempt

To be added.