
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
Suppose  $A=\{1,2,\dots,2002\}$  and  $M=\{1001,2003,3005\}$ .  $B$  is an non-empty subset of  $A$ .  $B$  is called a  $M$ -free set if the sum of any two numbers in  $B$  does not belong to  $M$ . If  $A=A_1\cup A_2$ ,  $A_1\cap A_2=\emptyset$  and  $A_1,A_2$  are  $M$ -free sets, we call the ordered pair  $(A_1,A_2)$  a  $M$ -partition of  $A$ . Find the number of  $M$ -partitions of  $A$ .

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