
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Let  $M=\{0,1,2,\dots,2022\}$  and let  $f:M\times M\to M$  such that for any  $a,b\in M$ ,
\[f(a,f(b,a))=b\]
and  $f(x,x)\neq x$  for each  $x\in M$ . How many possible functions  $f$  are there  $\pmod{1000}$ ?

Present the answer in LaTex format: \boxed{Your answer}
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