Orbital Period of an Asteroid What's the orbital period of an asteroid which has synodic period of 429 days?

I know that the formula of orbital period for inner planets is $$P_{orb} = \frac{P_{syn}}{P_{syn} + 1}$$ and for outer planets is $$P_{orb} = \frac{P_{syn}}{P_{syn} - 1}$$ where $P_{syn}$ is the synodic period of the planet measured in years. However, I don't know whether the asteroid should be counted as an inner planet or outer planet.
Just to clarify, inner planets usually refers to Mercury, Venus, Earth, Mars.    But in relation to your question, the "inner" planets are just Mercury and Venus and the outer planets are Mars and out, because the +1 / -1 formula is in relation to whether the Earth is passing the planet as it orbits the sun or being passed by the planet.   

Someone else can do the math if they like, the gist is explained here.  Because it's the motion relative to the stars in the background that you're tracking, anything that orbits closer to the sun and faster than Earth would move in a different direction and require a different formula than orbits further out and slower.

If your asteroid remains outside the Earth's orbit, then it would obey the outer planet formula.

If your asteroid is an Earth crosser, which isn't uncommon, then it would have a different type of orbital period that would follow neither formula, and I'm not sure what formula would apply for an earth-crossing asteroid.