9 was the best I could see for the first output and I found a 4-cycle pipeline, but there's one stutterstep I couldn't eliminate when I merge the two, the outside path for the water and salt takes one extra step as compared to the Animismus. After that one step is resolved then the main pipeline can potentially output continuously with a cycle of 4. Only that I can't set a repeating set of instructions on arm 10 due to the initialisation for the first output.

The second pipeline makes one output for every 10 pulls. After 20 inputs we can loop, the two pipelines end up on the same parity to allow for repeating instructions. But due to the reset needed for arm 10, it's slower than to just add more instructions. 
I can add any arbitrarily long set of instructions (of sets of 20 inputs) to make the Lexicographic order:
9 [5 4 4 4 4 2 2 [4 4 4 4 2 2 4 4 4 4 2 2] 4 4 4 7 2]
I programmed out 40 inputs to see what it looks like. Hopefully I don't lose a primary tie at output 24.