The relation for calculating the [synodic period][1] of two bodies is rather simple

> If the orbital periods of the two bodies around the third are called $T_1$ and $T_2$, so that $T_1 < T_2$, their synodic period is given by: $1 \over T_{syn}$$=$$1 \over T_1$$-$$1 \over T_2$

The [relationship between orbital radius and orbital period][2] is given by $T= 2\pi\sqrt{a^3/GM}$

Wolphram Alpha helps calculating that:

 - [10 million miles are][3] 0.1 au
 - the [orbital period at 1.1 au][4] would be 421.4 days
 - the [orbital period at 0.734 au][5] would be 229.7 days

therefore the synodic period, based on the formula above, would be 504.9 days


  [1]: https://en.wikipedia.org/wiki/Orbital_period#Synodic_period
  [2]: https://en.wikipedia.org/wiki/Orbital_period#Small_body_orbiting_a_central_body
  [3]: https://www.wolframalpha.com/input?i=10%20million%20miles%20to%20au
  [4]: https://www.wolframalpha.com/input?i=orbital%20period%20for%201.1%20au
  [5]: https://www.wolframalpha.com/input?i=orbital%20period%20for%200.734%20au