Define a Martin quadruple {a,b,c,d} as a solution in non-zero integers to the system,
$a+b+c+d = x^2$
$a^2+b^2+c^2+d^2 = y^2$
$a^3+b^3+c^3+d^3 = z^3$
It can be shown that there are an infinite number of solutions. However, the smallest five that are positive and 6th-power primitive (no common factor that is a 6th power, re cyclochaotic's comment below) have the curious linear sums as smooth numbers,
$\begin{aligned} &10 + 13 + 14 + 44 = 9^2 = 3^4\\ &54 + 109 + 202 + 260 = 25^2 = 5^4\\ &102 + 130 + 234 + 318 = 28^2 = 2^4\cdot7^2\\ &198 + 630 + 1594 + 1674 = 64^2 = 2^{12}\\ &570 + 742 + 1094 + 1690 = 64^2 = 2^{12}\end{aligned}$
found by James Allen and Seiji Tomita. Of course, the squares and the cubes of the addends also add up to a square and cube, respectively.
What is the sixth such quadruple?