
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  $n$  be a given positive integer. Let  $\mathbb{N}_+$  denote the set of all positive integers.

Determine the number of all finite lists  $(a_1,a_2,\cdots,a_m)$  such that:**(1)**  $m\in \mathbb{N}_+$  and  $a_1,a_2,\cdots,a_m\in \mathbb{N}_+$  and  $a_1+a_2+\cdots+a_m=n$ .**(2)** The number of all pairs of integers  $(i,j)$  satisfying  $1\leq i<j\leq m$   and  $a_i>a_j$  is even.

For example, when  $n=4$ , the number of all such lists  $(a_1,a_2,\cdots,a_m)$  is  $6$ , and these lists are  $(4),$   $(1,3),$   $(2,2),$   $(1,1,2),$   $(2,1,1),$   $(1,1,1,1)$ .

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