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Blend the strawberries then push the resulting puree through a sieve to remove some of the seeds. Tip the sieved puree into the blender again and add the ice, rum, 1-2 tbsp sugar syrup depending on how sweet you like your drinks and lime juice. Blend again and divide the mixture between 2 Martini glasses.
Thread the lime slices and strawberry halves onto the cocktail sticks and place onto the edge of the glass, serve immediately.