How can I find the Schmidt decomposition of the following state of two qubits
$$| \psi\rangle=\frac{|0\rangle|0\rangle+|0\rangle|1\rangle+|1\rangle|0\rangle}{\sqrt{3}} $$
i.e I want to find the ortonormal states $\{ |\alpha_{i}\rangle\}$ and $\{ |\beta_{i}\rangle\}$ such that
$$ |\psi\rangle=\sum_{i}\lambda_{i}|\alpha_{i}\rangle|\beta_{i}\rangle $$
where $\lambda_{i}$ are the Schmidt coefficients.
I proved to add and subtract the same terms to rearrange $|\psi\rangle$ into combinations of $|0\rangle + |1\rangle$ and $|0\rangle - |1\rangle$ but I don't find a coherent solution.
Note: $\langle0|0\rangle=\langle1|1\rangle=1$, $\langle0|1\rangle=0$, are two ortonormal vectors usually used in quantum mechanics.