Let \(\displaystyle S_n(x) = \log_{a^{1/2}} x + \log_{a^{1/3}} x + \log_{a^{1/6}} x + \log_{a^{1/11}} x + \log_{a^{1/18}} x + \log_{a^{1/27}} x + \dots\dots \) upto \( n \)-terms, where \( a>1 \). If \( S_{24}(x)=1093 \) and \( S_{12}(2x)=265 \), then value of \( a \) is equal to __________.
