
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 arithmetic sequence $\{a_n\}$ and a geometric sequence $\{b_n\}$ satisfying $a_1 + a_2 = a_3$, $b_1b_2 = b_3$, and $a_3$, $a_2 + b_1$, $a_1 + b_2$ form an arithmetic sequence, and $a_1$, $a_2$, $b_2$ form a geometric sequence.
(1) Find the general formula for sequences $\{a_n\}$ and $\{b_n\}$;
(2) Take terms from sequences $\{a_n\}$ and $\{b_n\}$ in the following manner:
The first time, take $a_1$ from sequence $\{a_n\}$,
The second time, take $b_1$, $b_2$ from sequence $\{b_n\}$,
The third time, take $a_2$, $a_3$, $a_4$ from sequence $\{a_n\}$,
The fourth time, take $b_3$, $b_4$, $b_5$, $b_6$ from sequence $\{b_n\}$,
...
The $(2n-1)$th time, continue to take $2n-1$ terms in order from sequence $\{a_n\}$,
The $2n$th time, continue to take $2n$ terms in order from sequence $\{b_n\}$,
...
Thus, construct sequence $\{c_n\}$: $a_1$, $b_1$, $b_2$, $a_2$, $a_3$, $a_4$, $b_3$, $b_4$, $b_5$, $b_6$, $a_5$, $a_6$, $a_7$, $a_8$, $a_9$, $b_7$, $b_8$, $b_9$, $b_{10}$, $b_{11}$, $b_{12}$, ..., let the sum of the first $n$ terms of sequence $\{c_n\}$ be $S_n$, find the largest positive integer $n$ such that $S_n < 2^{2014}$.

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