In \(\mathbb R^2\) if the magnitude of the projection vector of the vector \(\alpha \hat +\beta \hat j\) on \(\sqrt3 \hat i +\hat j\) is \(\sqrt3\) and if \(\alpha=2+\sqrt3 \beta\), then find the possible value(s) of \(|\alpha|\). \[\begin{array}{} (1) \, 1 \quad \quad \quad \quad \quad \quad \quad \quad & (2) \, 2 \\ (3) \, 3 & (4) \, 4 \end{array}\]

**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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