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The back end is squirrely for the first few miles in cooler temperatures (40s and 50s Fahrenheit) on the stock 220 treadwear Goodyear Supercar 3 tires if I go anywhere near full throttle even with traction control on. I can imagine a less grippy higher treadwear snow tire would be even worse for heavy acceleration even if it is labeled as "performance" but not sure on the oscillations.
If you have a safe flat wide open empty parking lot you could try holding the traction control button down for 10 seconds to fully disable it and do some back to back light acceleration testing at low/safe speeds to narrow down if it is the car's electronics causing the oscillation or possibly the physical properties of the tires. Disclaimer: I'm not responsible if you lose control with traction control off and don't try that with any other cars or objects nearby.
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