
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
The sequence \(\{a_{n}\}\) has consecutive terms \(a_{n}\) and \(a_{n-1}\) which are the roots of the equation \(x^{2} - c_{n} x + \left(\frac{1}{3}\right)^{n} = 0\), and \(a_{1} = 2\). Find the sum of the first \(2n\) terms of the sequence \(\{c_{n}\}\), denoted as \(S_{2n}\).

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