You will need a cooler with a lga1700 mounting kit.
Most coolers have announced such kits, either for free or for a modest charge.
Old stock may not have lga1700 mounting included.
The A400 is a 120mm cooler with 4 heat pipes.
The cooling is sufficient, up to a point.
But, if you are spending so much for a great processor like the 12600K,
You should consider a stronger cooler.
A good cooler is needed to allow the full turbo mechanism to take place.
If your room temperature is hot, that hot air is what the cooler gets to work with.
Noctua coolers now come with lga1700 mounting.
here is a list of suitable coolers for the 12600k:
The least expensive suitable cooler is likely to be the NH-U12S redux.
But, I would go stronger given your hot environment.
What is the make/model of your case?
Most good coolers will need perhaps 160mm headroom.