
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 all subsequences \( \{a_{1}, a_{2}, \cdots, a_{n}\} \) of the sequence \(\{1, 2, \cdots, n\}\) that satisfy the property \( i + 1 \mid 2(a_{1} + a_{2} + \cdots + a_{j}) \) for \(1 \leq i \leq n\).

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