Find the Missing Numbers in the Fibonacci Sequence Mod K Inspired by this Math.SE question.

Background

The Fibonacci Sequence (called F) is the sequence, starting 0, 1 such that each number (F(n)) (after the first two) is the sum of the two before it (F(n) = F(n-1) + F(n-2)).

A Fibonacci Sequence mod K (called M) is the sequence of the Fibonacci numbers mod K (M(n) = F(n) % K).

It can be shown that the Fibonacci Sequence mod K is cyclic for all K, as each value is determined by the previous pair, and there are only K2 possible pairs of non-negative integers both less than K. Because the Fibonacci sequence mod K is cyclic after its first repeated pair of terms, a number which doesn't appear in the Fibonacci Sequence mod K before the first repeated pair of terms will never appear.

For K = 4

0 1 1 2 3 1 0 1 ...

For K = 8

0 1 1 2 3 5 0 5 5 2 7 1 0 1 ...

Notice that for K = 8, 4 and 6 do not appear before the repeated 0 1, so 4 and 6 will never appear in the Fibonacci Sequence mod 8. 

Challenge

Given an integer K strictly greater than 0, output all of the non-negative integers less than K that do not appear in the Fibonacci Sequence mod K.

Rules


Default loopholes are forbidden.
Default I/O.
Programs or functions are acceptable.
You can assume that K will fit in your native integer type (within reason).
If there are non-negative numbers less than K that do not appear in the Fibonacci Sequence mod K, your program/function should output all such numbers in any reasonable manner.
If there are no non-negative integers less than K that do not appear in the Fibonacci Sequence mod K, your program/function may indicate this by returning an empty list, printing nothing, producing an error, etc.
Order does not matter.
This is code-golf, so shortest answer in each language wins.


Test Cases

Generate test cases online!

Non-Empty Test Cases

  8 [4, 6]
 11 [4, 6, 7, 9]
 12 [6]
 13 [4, 6, 7, 9]
 16 [4, 6, 10, 12, 14]
 17 [6, 7, 10, 11]
 18 [4, 6, 7, 9, 11, 12, 14]
 19 [4, 6, 7, 9, 10, 12, 14]
 21 [4, 6, 7, 9, 10, 11, 12, 14, 15, 16, 17, 19]
 22 [4, 6, 7, 9, 15, 17, 18, 20]
 23 [4, 7, 16, 19]
 24 [4, 6, 9, 11, 12, 14, 15, 18, 19, 20, 22]
 26 [4, 6, 7, 9, 17, 19, 20, 22]
 28 [10, 12, 14, 16, 18, 19, 23]
 29 [4, 6, 7, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 27]
 31 [4, 6, 9, 12, 14, 15, 17, 18, 19, 22, 25, 29]
 32 [4, 6, 10, 12, 14, 18, 20, 22, 26, 28, 30]
 33 [4, 6, 7, 9, 15, 17, 18, 20, 24, 26, 27, 28, 29, 31]
 34 [4, 6, 7, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 27, 28, 30]
 36 [4, 6, 7, 9, 10, 11, 12, 14, 16, 18, 20, 22, 23, 24, 25, 26, 27, 29, 30, 31, 32]
 37 [9, 10, 14, 17, 20, 23, 27, 28]
 38 [4, 6, 7, 9, 10, 11, 12, 14, 15, 16, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 36]
 39 [4, 6, 7, 9, 15, 17, 19, 20, 22, 24, 30, 32, 33, 35]
...
200 [4, 6, 12, 14, 20, 22, 28, 30, 36, 38, 44, 46, 52, 54, 60, 62, 68, 70, 76, 78, 84, 86, 92, 94, 100, 102, 108, 110, 116, 118, 124, 126, 132, 134, 140, 142, 148, 150, 156, 158, 164, 166, 172, 174, 180, 182, 188, 190, 196, 198]
...
300 [6, 18, 30, 42, 54, 66, 78, 90, 102, 114, 126, 138, 150, 162, 174, 186, 198, 210, 222, 234, 246, 258, 270, 282, 294]
...
400 [4, 6, 10, 12, 14, 20, 22, 26, 28, 30, 36, 38, 42, 44, 46, 52, 54, 58, 60, 62, 68, 70, 74, 76, 78, 84, 86, 90, 92, 94, 100, 102, 106, 108, 110, 116, 118, 122, 124, 126, 132, 134, 138, 140, 142, 148, 150, 154, 156, 158, 164, 166, 170, 172, 174, 180, 182, 186, 188, 190, 196, 198, 202, 204, 206, 212, 214, 218, 220, 222, 228, 230, 234, 236, 238, 244, 246, 250, 252, 254, 260, 262, 266, 268, 270, 276, 278, 282, 284, 286, 292, 294, 298, 300, 302, 308, 310, 314, 316, 318, 324, 326, 330, 332, 334, 340, 342, 346, 348, 350, 356, 358, 362, 364, 366, 372, 374, 378, 380, 382, 388, 390, 394, 396, 398]
...


Empty Test Cases (no output, error, empty list, etc. is acceptable output)

1, 2, 3, 4, 5, 6, 7, 9, 10, 14, 15, 20, 25, 27, 30, 35 ... 100 ...


Related:

Counting Fibonacci Orbits

Find the Pisano Period
Haskell, 70 bytes

Some amount of bytes saved thanks to Esolanging Fruit

8 bytes saved thanks to Laikoni



a=1:scanl(+)1a
f x=[u|u&lt;-[2..x-1],and[mod b x/=u|(_,b)&lt;-zip[1..x^2]a]]


Try it online!