There are 14 levels in FACE.
I don't know the actual enumeration of the levels as the achievements pop up after the game is closed, but it isn't really hard to visit them all and you can always deduce the numbers if you already know some of them.
The starting level is level 1 in my guide.

There is a schematic of the doors at the end.

Level 1:

• Left door leads to level 2
  
• Right door leads to level 5

Level 2:

• Left door leads to level 3
  
• Right door leads to level 1

Level 3:

• Left door leads to level 2
  
• Right door leads to level 4

Level 4:

• Left door leads to level 3
  
• Middle door leads to level 7
  
• Right door leads to level 5

Level 5:

• Left door leads to level 1
  
• Middle door leads to level 4
  
• Right door leads to level 6

Level 6:

• Left door leads to level 5
  
• Right door leads to level 7

Level 7:

• Left door leads to level 8
  
• Middle door leads to level 4
  
• Right door leads to level 6

Level 8:

• Left door leads to level 7
  
• Right door leads to level 9

Level 9:

• Left door leads to level 12
  
• Middle door leads to level 8
  
• Right door leads to level 10

Level 10:

• Left door leads to level 11
  
• Right door leads to level 9

Level 11:

• Left door leads to level 12
  
• Middle door leads to level 10
  
• Right door leads to level 14

Level 12:

• Upper left door leads to level 9
  
• Lower left door leads to level 11
  
• Right door leads to level 13

Level 13:

• Left door leads to level 12
  
• Right door leads to level 14

Level 14:

• Left door leads to level 11
  
• Right door leads to level 13