Question #40565

Let $S$ be the set of all distinct permutations of the multiset $\{1, 2, 3\}$. For each permutation $\sigma = (\sigma_1, \sigma_2, \sigma_3) \in S$, a plane $P_\sigma$ is defined by the equation $\frac{x}{\sigma_1} + \frac{y}{\sigma_2} + \frac{z}{\sigma_3} = 1$. Let $Q$ be the unique point of intersection of all the planes $P_\sigma$ for $\sigma \in S$. Let $N_1, N_2,$ and $N_3$ be the number of distinct permutations of the letters of the words 'TO', 'JEE', and 'NOON' respectively. A plane $\Pi$ contains the origin $O$, the point $Q$, and the point $R(N_1, N_2, N_3)$. A line $L$ lies in the plane $\Pi$, passes through the origin, and is perpendicular to the line $OQ$. If the direction ratios of $L$ are $(a, b, c)$ where $a, b, c$ are coprime integers and $a > 0$, find the value of $a^2 + b^2 + c^2$.