Synthetic key signatures with double accidentals

In some contemporary music, you run into synthetic scales with double sharps and flats.
If you need to display the synthetic key signature, you can even let DOUBLE-SHARP and DOUBLE-FLAT be printed.

\version "2.24"

synthetic = 
  #`((0 . ,NATURAL) 
     (1 . ,DOUBLE-SHARP) 
     (2 . ,SHARP)
     (3 . ,NATURAL) 
     (4 . ,FLAT) 
     (5 . ,FLAT) 
     (6 . ,DOUBLE-FLAT))

m = \relative c' { \omit Staff.TimeSignature c8 disis eis f ges aes beses c }

<<
  \new Staff \m
  \new Staff { \key c \synthetic \m }
>>