Coffee machine buttons - how to connect a relay?

Hi there! I'm trying to connect a relay to two buttons of a coffee machine to control it remotely.
When I opened it up I discovered that the two buttons are placed on a separate mini circuit board.
As a more or less total newbie the layout confuses me a bit. I tried to determine in which way a button press (small cup, large cup) influences the current flow using a multimeter but I still do not know how to connect a relay correctly.
If someone could give me a hint and maybe also accomplishes to explain me what the white squares are for I'd be truly grateful!

I've attached two photos (front,back) and three sketches of the currency flow without any button pressed, the changes when the left button is pressed and the changes when the right button is pressed.

Thanks in advance!

Are the buttons on this board, or somewhere else?
It appears several contacts are un-made with each button press?
You may need a relay with several sets of contacts, with the Normally Closed connections the default state, and going to Open when the relay energizes.

Warning: using high voltage/ high current can be dangerous. A mistake could cause personal injury or fires.
Proceed at your own risk.

I assume these buttons are push buttons, normally open, and closed only as long as you press them. Is that correct. Have you checked them with a volt-ohm meter (carefully)?

If those assumptions are accurate, how do you think you should hook a relay up? Draw us a schematic.