
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
For the sequence \( \{a_{n}\} \), the two adjacent terms \( a_{n} \) and \( a_{n+1} \) are the roots of the quadratic equation \( x^{2} - 3nx + c_{n} = 0 \) (for \( n = 1, 2, 3, \ldots \)). Given that \( a_{1} = 1 \), find \( c_{n} \).

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