
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
In a hidden friend, suppose no one takes oneself. We say that the hidden friend has "marmalade" if
there are two people  $A$  and  $ B$  such that A took  $B$  and  $ B $  took  $A$ . For each positive integer n, let  $f (n)$  be the number of hidden friends with n people where there is no “marmalade”, i.e.  $f (n)$  is equal to the number of permutations  $\sigma$  of { $1, 2,. . . , n$ } such that:

* $\sigma (i) \neq i$  for all  $i=1,2,...,n$ * there are no  $ 1 \leq i <j \leq n $  such that  $ \sigma (i) = j$  and  $\sigma (j) = i. $ Determine the limit

  $\lim_{n \to + \infty} \frac{f(n)}{n!}$ 

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