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\includegraphics[max width=\textwidth, alt={}, center]{616a6177-0d5c-49f7-b0c1-9138a13c1963-2_663_446_1562_847} In the diagram, triangle \(A B C\) is right-angled at \(C\) and \(M\) is the mid-point of \(B C\). It is given that angle \(A B C = \frac { 1 } { 3 } \pi\) radians and angle \(B A M = \theta\) radians. Denoting the lengths of \(B M\) and \(M C\) by \(x\),

1. find \(A M\) in terms of \(x\),
2. show that \(\theta = \frac { 1 } { 6 } \pi - \tan ^ { - 1 } \left( \frac { 1 } { 2 \sqrt { 3 } } \right)\).