
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  $n$  ( $n\ge 3$ ) be a natural number. Denote by  $f(n)$  the least natural number by which  $n$  is not divisible (e.g.  $f(12)=5$ ). If  $f(n)\ge 3$ , we may have  $f(f(n))$  in the same way. Similarly, if  $f(f(n))\ge 3$ , we may have  $f(f(f(n)))$ , and so on. If  $\underbrace{f(f(\dots f}_{k\text{ times}}(n)\dots ))=2$ , we call  $k$  the “*length*” of  $n$  (also we denote by  $l_n$  the “*length*” of  $n$ ). For arbitrary natural number  $n$  ( $n\ge 3$ ), find  $l_n$  with proof.

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