
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 sequence \\(\\{a\_{n}\\}\\) that satisfies: \\(a\_{1}+2a\_{2}+2^{2}a\_{3}+…+2^{n-1}a\_{n}= \frac{n^{2}+1}{3}(n\in\mathbb{N}^{*})\\), find the value of \\(a\_{n}\\).

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