
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
From the set of all permutations  $f$  of  $\{1, 2, ... , n\}$  that satisfy the condition:
 $f(i) \geq i-1$     $i=1,...,n$ 
one is chosen uniformly at random. Let  $p_n$  be the probability that the chosen permutation  $f$  satisfies
 $f(i) \leq i+1$    $i=1,...,n$ 
Find all natural numbers  $n$  such that  $p_n > \frac{1}{3}$ .

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