Say no to Azenis for snow and heavy rain!!!!!! :nono::nono::nono::nono:
Speaking from personal experience since I've had these tires on my car for the past year, these are not what you are looking for as an "all-season" tire.
What makes them such a demanding tire for dry, track use, is the fact that there is a ton of tread and not much of anything else (water channels, grooves, etc.). The reason why they're recommended for dry weather use because they heat up quick, grip exceptionally well, and last you a decent while.
The trade-off is that with so much tread block, there's no water channels to allow water to escape from underneath the tire which can cause hydroplaning (I've done that before) and hardly any grooves (tread pattern) with which to grip the snow and gain traction (been there too).
What helps the tire perform in dry weather (heat buildup, therefore a sticky tire) cannot happen in cold or wet weather so the tire is useless and can possibly send you into an accident.
If you're looking for tires that can handle the snow, look for some with the designations M+S, M/S, M&S, or MS. This indicates that these pass RMA (Rubber Manufacturers Association) guidelines for Mud and Snow tires.