
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  $f$  be a function from the set  $X = \{1,2, \dots, 10\}$  to itself. Call a partition ( $S$ ,  $T$ ) of  $X$   $f-balanced$  if for all  $s \in S$  we have  $f(s) \in T$  and for all  $t \in T$  we have  $f(t) \in S$ . (A partition  $(S, T)$  is a pair of subsets  $S$  and  $T$  of  $X$  such that  $S\cap T = \emptyset$  and  $S \cup T = X$ . Note that  $(S, T)$  and  $(T, S)$  are considered the same partition).
    
Let  $g(f)$  be the number of  $f-balanced$  partitions, and let  $m$  equal the maximum value of  $g$  over all functions  $f$  from  $X$  to itself. If there are  $k$  functions satisfying  $g(f) = m$ , determine  $m+k$ .

*2016 CCA Math Bonanza Individual #12*

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