
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  $d(n)$  be the number of positive divisors of a positive integer  $n$ .  Let  $\mathbb{N}$  be the set of all positive integers.  Say that a function  $F$  from  $\mathbb{N}$  to  $\mathbb{N}$  is *divisor-respecting* if  $d(F(mn)) = d(F(m)) d(F(n))$  for all positive integers  $m$  and  $n$ , and  $d(F(n)) \le d(n)$  for all positive integers  $n$ .  Find all divisor-respecting functions.  Justify your answer.

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