Let \(N\) denote the number that turns up when a fair die is rolled. If the probability that the system of equations \(\…

Let \(N\) denote the number that turns up when a fair die is rolled. If the probability that the system of equations
\(\begin{aligned}& x+y+z=1 \\& 2 x+N y+2 z=2 \\& 3 x+3 y+N z=3\end{aligned}\)
has unique solution is \(\frac{k}{6}\), then the sum of value of \(k\) and all possible values of \(N\) is

[JEE Main 2023, 24 Jan (Shift 1)]