New iPhone & I already have an annual AppleTV+ sub. How does the 3 month free offer work on top of a current sub?

I already pay for a yearly sub for AppleTV+, but I purchased 2 new iPhone 14 phones and keep getting prompts to get it free for 3 months. However, it seems as though if I do this, my monthly price will go up to $7.99 a month. Not sure if this will give me a second subscription, or cancel out my current one (which it seems it doesn't see) or screw me on the price going forward. Anyone know what's up with this?