
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
Consider a natural number  $ n $  for which it exist a natural number  $ k $  and  $ k $  distinct primes so that  $ n=p_1\cdot p_2\cdots p_k. $ **a)** Find the number of functions  $ f:\{ 1, 2,\ldots , n\}\longrightarrow\{ 1,2,\ldots ,n\} $  that have the property that  $ f(1)\cdot f(2)\cdots f\left( n \right) $  divides  $ n. $ **b)** If  $ n=6, $  find the number of functions  $ f:\{ 1, 2,3,4,5,6\}\longrightarrow\{ 1,2,3,4,5,6\} $  that have the property that  $ f(1)\cdot f(2)\cdot f(3)\cdot f(4)\cdot f(5)\cdot f(6) $  divides  $ 36. $ 

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