
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
A set  $P$  consists of  $2005$  distinct prime numbers. Let  $A$  be the set of all possible products of  $1002$  elements of  $P$  , and  $B$  be the set of all products of  $1003$  elements of  $P$  . Find a one-to-one correspondance  $f$  from  $A$  to  $B$  with the property that  $a$  divides  $f (a)$  for all  $a \in A.$ 

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