
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
The set $\mathrm{X}=\{1,2, \ldots, 100\}$ and the function $\mathrm{f}: \mathrm{X} \rightarrow \mathrm{X}$ satisfy the following conditions:  
(1) For any $\mathrm{x} \in \mathrm{X}$, $f(x) \neq x$;  
(2) For any subset $A$ of $X$ with 40 elements, $A \cap f(A) \neq \emptyset$.  

Find the smallest positive integer $\mathrm{k}$ such that for any function $\mathrm{f}$ that satisfies these conditions, there exists a $\mathrm{k}$-element subset $B$ of $\mathrm{X}$ such that $B \cup f(B) = X$. (Note: For a subset $T$ of $X$, define $f(T)=\{x \mid x=f(t), t \in T\}$.)

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