Click and hold the line segment tool. Select the spiral tool.
1) First outer curve: Number of segments: 3. Adjust to fit. Rotate a copy 180 degrees round the center of the circle. 2) Inner curve: Number of segments: 2 Adjust to fit outer curve. Rotate a copy 180 degrees round the centre of the circle. 3) Join paths. 4) Make compound path with circle.
Draw one of those parts (using the pen tool) and then duplicate it. Since the template is not even symmetric, you will have to do some construction work with this. You need to improve it as you work along.