<SYSTEM_TASK:>
Solve the following problem using Python, implementing the functions described below, one line at a time
<END_TASK>
<USER_TASK:>
Description:
 def guess_mime_type(self, path):
 		""" 		Guess an appropriate MIME type based on the extension of the 		provided path. 		:param str path: The of the file to analyze. 		:return: The guessed MIME type of the default if non are found. 		:rtype: str 		"""

		_, ext = posixpath.splitext(path)
		if ext in self.extensions_map:
			return self.extensions_map[ext]
		ext = ext.lower()
		return self.extensions_map[ext if ext in self.extensions_map else '']