\[\begin{array}{l} \angle BMD = \angle BND = \frac{\pi }{3} \Rightarrow \angle QSD = \frac{\pi }{{12}} \hfill \\ \frac{{BD}}{{\sin \frac{\pi }{3}}} = 2R \Leftrightarrow R = \sqrt {\frac{2}{3}} \hfill \\ MC = \frac{{\sqrt 3 }}{3} \hfill \\ \frac{{QD}}{{\sin \angle QSD}} = 2R \Leftrightarrow QD = BS = 1 - \frac{{\sqrt 3 }}{3} \hfill \\ SC = \sqrt {B{S^2} + B{C^2}} = \frac{{\sqrt {21 - 6\sqrt 3 } }}{3} \hfill \\ PC \cdot SC = MC \cdot CD \Leftrightarrow PC = \frac{1}{{\sqrt {7 - 2\sqrt 3 } }} \hfill \\ x = SP = SC + PC = \frac{{\sqrt {21 - 6\sqrt 3 } }}{3} + \frac{1}{{\sqrt {7 - 2\sqrt 3 } }} = \frac{{7\sqrt 3 - 3}}{{3\sqrt {7 - 2\sqrt 3 } }}. \hfill \\ \end{array}\]

\[\frac{{7\sqrt 3 - 3}}{{3\sqrt {7 - 2\sqrt 3 } }}\]