
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 an integer \( k \geq 2 \). Find all positive integers \( n_{1}, n_{2}, \cdots, n_{k} \) that satisfy:

\[ 
\begin{array}{l}
n_{2} \mid \left(2^{n_{1}} - 1\right), n_{3} \mid \left(2^{n_{2}} - 1\right), \cdots, \\
n_{k} \mid \left(2^{n_{k-1}} - 1\right), n_{1} \mid \left(2^{n_{k}} - 1\right).
\end{array}
\]

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