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The iPhone 2 Bluetooth headset can be charged with an iPhone 13 cable. Most of Apple's products are generic and will have no impact.
For the first time, try to make the headset charged for 2-4 hours or so, after 2 hours can be charged, the manual also says so, it is best not to exceed 4 hours, some forums say that must be charged for 24 hours is unfounded, such a small headset, charged for 24 hours, some too exaggerated.