Find all possible sextuplets (x,y,z,p,q,r) of positive integers that satisfy this system of equations:

• x⁴+14xy+1 = p⁴
• y⁴+14yz+1 = q⁴
• z⁴+14xz+1 = r⁴

Prove that there are no further valid sextuplets satsfying the given conditions.

Notes:
• Adapted from a problem appearing in Soros Math Olympiad.
• Computer program assisted solutions are welcome, but a semi-analytic solution is preferred.