
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
For an ordered pair  $(m,n)$  of distinct positive integers, suppose, for some nonempty subset  $S$  of  $\mathbb R$ , that a function  $f:S \rightarrow S$  satisfies the property that  $f^m(x) + f^n(y) = x+y$  for all  $x,y\in S$ . (Here  $f^k(z)$  means the result when  $f$  is applied  $k$  times to  $z$ ; for example,  $f^1(z)=f(z)$  and  $f^3(z)=f(f(f(z)))$ .) Then  $f$  is called \emph{ $(m,n)$ -splendid}. Furthermore,  $f$  is called \emph{ $(m,n)$ -primitive} if  $f$  is  $(m,n)$ -splendid and there do not exist positive integers  $a\le m$  and  $b\le n$  with  $(a,b)\neq (m,n)$  and  $a \neq b$  such that  $f$  is also  $(a,b)$ -splendid. Compute the number of ordered pairs  $(m,n)$  of distinct positive integers less than  $10000$  such that there exists a nonempty subset  $S$  of  $\mathbb R$  such that there exists an  $(m,n)$ -primitive function  $f: S \rightarrow S$ .

*Proposed by Vincent Huang*

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