Optimization & Lagrange Multipliers · Superchallenge problem

Four points $P_1, P_2, P_3, P_4$ lie on the unit sphere $x^2 + y^2 + z^2 = 1$. Find the largest possible value of the sum of the squares of the six distances $|P_iP_j|$, $1 \le i < j \le 4$.