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Solve the following problem using Python, implementing the functions described below, one line at a time
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Description:

def _update_schema_to_aws_notation(self, schema):
        '''
        Helper function to map model schema to aws notation
        '''
        result = {}
        for k, v in schema.items():
            if k == '$ref':
                v = self._aws_model_ref_from_swagger_ref(v)
            if isinstance(v, dict):
                v = self._update_schema_to_aws_notation(v)
            result[k] = v
        return result