
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 a positive integer \( n \), a finite sequence of positive integers \( a_{1}, a_{2}, \cdots, a_{m} \) is called an "n-even sequence" if \( n = a_{1} + a_{2} + \cdots + a_{m} \) and there are an even number of integer pairs \( (i, j) \) satisfying \( 1 \leq i < j \leq m \) and \( a_{i} > a_{j} \). Find the number of n-even sequences. For example, there are six 4-even sequences: \( 4 ; 1,3 ; 2,2 ; 1,1,2 ; 2,1,1 ; 1,1,1,1 \).

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