
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
For any number  $n\in\mathbb{N},n\ge 2$ , denote by  $P(n)$  the number of pairs  $(a,b)$  whose elements are of positive integers such that 
\[\frac{n}{a}\in (0,1),\quad \frac{a}{b}\in (1,2)\quad \text{and}\quad \frac{b}{n}\in (2,3). \] $a)$  Calculate  $P(3)$ . $b)$  Find  $n$  such that  $P(n)=2002$ .

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