
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
Le  $S$  be the set of positive integers greater than or equal to  $2$ . A function  $f: S\rightarrow S$  is italian if  $f$  satifies all the following three conditions:
1)  $f$  is surjective 
2)  $f$  is increasing in the prime numbers(that is, if  $p_1<p_2$  are prime numbers, then  $f(p_1)<f(p_2)$ )
3) For every  $n\in S$  the number  $f(n)$  is the product of  $f(p)$ , where  $p$  varies among all the primes which divide  $n$  (For instance,  $f(360)=f(2^3\cdot 3^2\cdot 5)=f(2)\cdot f(3)\cdot f(5)$ ).
Determine the maximum and the minimum possible value of  $f(2020)$ , when  $f$  varies among all italian functions.

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