First off, reaction is under acidic conditions so you want to avoid negatively charged intermediates if possible. And to make matters worse, one of your intermediates contains charge seperation. Not to say that these things are absolutely impossible, but thinking about this problem there is definitely a more reasonable solution.
The bromine radical idea is non sensical immediately because you don't have a suitable bond to generate a radical. Bromine radical is generated from Br2 or NBS, neither of which are present in the problem.
You are on the right track with the SN1 type mechanism, but we just don't need such a strong nucleophile for the reaction to go. Is that a good hint?