
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

Given the set \( M = \{1, 2, \cdots, 2n\} \), where \( A = \{a_1, a_2, \cdots, a_n\} \) and \( B = \{b_1, b_2, \cdots, b_n\} \) are \( n \)-element subsets such that:
1. \( A \cup B = M \)
2. \( A \cap B = \varnothing \)
3. \( \sum_{k=1}^{n} a_k = \sum_{k=1}^{n} b_k \)

We call \( A \cup B \) an "equal-sum partition" of \( M \) (with \( A \cup B \) and \( B \cup A \) considered the same partition).

Determine the number of "equal-sum partitions" for the set \( M = \{1, 2, \cdots, 12\} \).

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