Actually it’s the other way around you need to go to the Jamaica consulate get a sealed letter which you take to a police precinct then return to the passport office with statement from police and birth certificate and passport application
Report it lost to the police station and they will advice you to go from there
First you would have to report the passport lost/stolen to police department ( in which ever country/state you believed the passport was lost/stolen eg. Mandeville, Jamaica or Brooklyn, New York). 2) Get the police report from the police department then, take the lost/stolen report along with the newly filled passport form and necessary documents to the passport office that serves you in the USA. 3) If everything is done properly then their shouldn't be an issue, you'll get your passport within 4-6weeks. Good luck
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