In \(\mathbb{R}^3\), consider the planes \(P_1:y=0\) and \(P_2:x+z=1\). Let \(P_3\) be a plane, different from \(P_1\) and \(P_2\), which passes through the intersection of \(P_1\) and \(P_2\). If the distance of the point \((0,1,0)\) from \(P_3\) is \(1\) and the distance of a point \((\alpha,\beta,\gamma)\) from \(P_3\) is \(2\), then which of the following relations is(are) true ?
\[\begin{array}{} (1) \, 2\alpha+\beta+2\gamma+2=0 \quad \quad \quad \quad & (2) \, 2\alpha-\beta+2\gamma+4=0 \\ (3) \, 2\alpha+\beta-2\gamma-10=0 & (4) \, 2\alpha-\beta+2\gamma-8=0 \end{array}\]

**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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