Let $A$ be a $2\times 2$ real matrix with entries from $\left\{0, 1\right\}$ and $\left|A\right|\ne 0$. Consider the following two statements;

$\left(P\right)$ If $A\ne I_2$, then $\left|A\right|=-1$

$\left(Q\right)$ If $\left|A\right|=1$, then $tr\left(A\right)=2$

Where $I_2$ denotes $2\times 2$ identity matrix and $tr\left(A\right)$ denotes the sum of the diagonal entries of $A$. Then

Let $S$ be the set of all $\lambda \in R$ for which the system of linear equations

$2x-y+2z=2$

$x-2y+\lambda z=-4$

$x+\lambda y+z=4$

has no solution. Then the set $S$

Let $a, b, c\in R$ be all non-zero and satisfies $a^3+b^3+c^3=2$. If the matrix $A=\left[\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right]\begin{array}{}\\ \\ \end{array}$satisfies $A^{T}A=I,$ then a value of $abc$ can be

Let $S$ be the set of all integer solutions $\left(x, y, z\right)$ of the system of equations

$x-2 y+5 z=0$

$-2 x+4 y+z=0$

$-7 x+14 y+9 z=0$

such that $15\le x^2+y^2+z^2\le 150$. Then, the number of elements in the set $S$ is equal to ..........