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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 var_expand(self, cmd, depth=0, formatter=DollarFormatter()):
 """Expand python variables in a string. The depth argument indicates how many frames above the caller should be walked to look for the local namespace where to expand variables. The global namespace for expansion is always the user's interactive namespace. """

        ns = self.user_ns.copy()
        try:
            frame = sys._getframe(depth + 1)
        except ValueError:
            # This is thrown if there aren't that many frames on the stack,
            # e.g. if a script called run_line_magic() directly.
            pass
        else:
            ns.update(frame.f_locals)

        try:
            # We have to use .vformat() here, because 'self' is a valid and common
            # name, and expanding **ns for .format() would make it collide with
            # the 'self' argument of the method.
            cmd = formatter.vformat(cmd, args=[], kwargs=ns)
        except Exception:
            # if formatter couldn't format, just let it go untransformed
            pass
        return cmd