
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
Find the number of partitions of the set \(\{1,2, \ldots, n\}\) into three subsets \(A_{1}, A_{2}, A_{3}\), some of which may be empty, such that the following conditions are satisfied:
1. After the elements of every subset have been put in ascending order, every two consecutive elements of any subset have different parity.
2. If \(A_{1}, A_{2}, A_{3}\) are all nonempty, then in exactly one of them the minimal number is even.

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