Googling the problem leads us to a [similar Quora post](https://www.quora.com/How-do-you-find-the-positive-integer-solutions-to-frac-x-y+z-+-frac-y-z+x-+-frac-z-x+y-4/answer/Alon-Amit) on this challenge.
Modifying the script in the Quora post for N=10, we can solve for a,b, and c.
To make all numbers positive, we can brute force a multiple of P until all 3 numbers are positive.
Here's the script in Sage:
ee = EllipticCurve([0,517,0,416,0])
P = ee(-416,4160)
for i in range(1,1000):
u = orig(P*i,10)
if a > 0 and b > 0 and c > 0: