Geometry Level 4

In $$\mathbb{R}^3$$, let $$L$$ be a straight line passing through the origin. Suppose that all the points on $$L$$ are at a constant distance from the two planes $$P_1:x+2y-z+1=0$$ and $$P_2:2x-y+z-1=0$$. Let $$M$$ be the locus of the feet of the perpendiculars drawn fro the points on $$L$$ to the plane $$P_1$$. Which of the following points lie(s) on $$M$$ ?
$\begin{array}{} (1) \, \left(0,-\frac{5}{6},-\frac{2}{3} \right) \quad \quad \quad \quad & (2) \, \left(-\frac{1}{6},-\frac{1}{3},\frac{1}{6} \right) \\ (3) \, \left( -\frac{5}{6},0,\frac{1}{6} \right) & (4) \, \left( -\frac{1}{3},0,\frac{2}{3} \right) \end{array}$
Note :

• Submit your answer as the increasing order of the serial numbers of all the correct options.

• For eg, if your answer is $$(1),(2)$$, then submit 12 as the correct answer, if your answer is $$(2),(3),(4)$$, then submit 234 as the correct answer.

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