Protein Shake Projekt (vertikal movment/Elivator)

Hello there,

(making it short)
Im new to Arduino, but have some Programming skills.

The PROJEKT

I have a projekt in my head.
Its a Automated Proteinshake maker.

The IDEA

-Using a Waterpump motor, to fill a cup of glass standing underneath.
-Putting in Protein powder using a big Screw like this:
https://www.amazon.de/dp/B06X9CSBT4/ref=cm_sw_em_r_mt_dp_U_V9fhEbFJMTSC9
-Then mixing (Problem)

Now comes the Problem:
I want the "mix wand" to come down a little into the glass, then mix, and lift up again.

by mix wand i mean: (but just the front part)
https://www.amazon.de/dp/B009MP7VY2/ref=cm_sw_em_r_mt_dp_U_qkghEbTBVP5S4

OWN SOLUTION Ideas

-using a cogwheel rail and a stepper motor (Problem is i cant imagine how the cogwheel rail only movies Vertikal, i mean how can i make its position fix, but still be able to move up and down.

  • using a toothed belt, and fixing the engine for the mix wand on it.

Does anyone know what i mean?

Does anybody have simpler solution?

In all i need (roughly)
-arduino
-waterpump
-1 motor for the mix wand
-1 stepper motor for the Protein powder
-1 stepper motor for the mix wand to move vertikal (Probably)

Heres a little sketch:
Imgur

I would be glad for any Help/advice
Thank you