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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 _create_alternative_resolutions(self, source_image, geometry_string, options, name):
 """ Creates the thumbnail by using default.engine with multiple output sizes. Appends @<ratio>x to the file name. """

        ratio = default.engine.get_image_ratio(source_image, options)
        geometry = parse_geometry(geometry_string, ratio)
        file_name, dot_file_ext = os.path.splitext(name)

        for resolution in settings.THUMBNAIL_ALTERNATIVE_RESOLUTIONS:
            resolution_geometry = (int(geometry[0] * resolution), int(geometry[1] * resolution))
            resolution_options = options.copy()
            if 'crop' in options and isinstance(options['crop'], string_types):
                crop = options['crop'].split(" ")
                for i in range(len(crop)):
                    s = re.match(r"(\d+)px", crop[i])
                    if s:
                        crop[i] = "%spx" % int(int(s.group(1)) * resolution)
                resolution_options['crop'] = " ".join(crop)

            image = default.engine.create(source_image, resolution_geometry, options)
            thumbnail_name = '%(file_name)s%(suffix)s%(file_ext)s' % {
                'file_name': file_name,
                'suffix': '@%sx' % resolution,
                'file_ext': dot_file_ext
            }
            thumbnail = ImageFile(thumbnail_name, default.storage)
            default.engine.write(image, resolution_options, thumbnail)
            size = default.engine.get_image_size(image)
            thumbnail.set_size(size)