
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
In the geometric sequence $\{a_n\}$, where the common ratio $q \neq 1$, there exists an arithmetic sequence $\{b_n\}$ satisfying $b_1 = a_1 = 3$, $b_4 = a_2$, and $b_{13} = a_3$.  
(I) Find the general formulae for the sequences $\{a_n\}$ and $\{b_n\}$.  
(II) Let $c_n = (-1)^n b_n + a_n$. Calculate the sum $S_{2n}$ of the first $2n$ terms of the sequence $\{c_n\}$.

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