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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