
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]


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Given a sequence $\{a_{n}\}$ that satisfies ${a}_{1}+3{a}_{2}+9{a}_{3}+⋯+{3}^{n-1}{a}_{n}=\frac{n+1}{3}$, where the sum of the first $n$ terms of the sequence $\{a_{n}\}$ is denoted as $S_{n}$, find the minimum value of the real number $k$ such that $S_{n} \lt k$ holds for all $n$.

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