Trapezoid \(ABCD\) has \(AD \parallel BC\), \(BD = 1\), \(\angle DBA = 23^\circ\), and \(\angle BDC = 46^\circ\). The ratio \(BC: AD\) is \(9: 5\). What is the value of \(CD\)? The original answer is in the form \(\frac{k}{m}\), where \(\frac{k}{m}\) is a fully simplified fraction. Please provide the value of \(k + m\).

Let's think step by step and output the final answer within \boxed{}.
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