The relation for calculating the synodic period 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 is given by $T= 2\pi\sqrt{a^3/GM}$
Wolphram Alpha helps calculating that:
- 10 million miles are 0.1 au
- the orbital period at 1.1 au would be 421.4 days
- the orbital period at 0.824 au would be 273.2 days
therefore the synodic period, based on the formula above, would be 776.8 days