
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  $f:X\rightarrow X$ , where  $X=\{1,2,\ldots ,100\}$ , be a function satisfying:
1)  $f(x)\neq x$  for all  $x=1,2,\ldots,100$ ;
2) for any subset  $A$  of  $X$  such that  $|A|=40$ , we have  $A\cap f(A)\neq\emptyset$ .
Find the minimum  $k$  such that for any such function  $f$ , there exist a subset  $B$  of  $X$ , where  $|B|=k$ , such that  $B\cup f(B)=X$ .

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