
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 $A = \{1, 2, \cdots, 2002\}$ and $M = \{1001, 2003, 3005\}$. For any non-empty subset $B$ of $A$, if the sum of any two elements in $B$ is not in $M$, then $B$ is called an $M$-free set. If $A = A_{1} \cup A_{2}$, $A_{1} \cap A_{2} = \emptyset$, and both $A_{1}$ and $A_{2}$ are $M$-free sets, then the ordered pair $\left(A_{1}, A_{2}\right)$ is called an $M$-partition of $A$. Find the number of all $M$-partitions of $A$.

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