
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 two sequences $\{a_n\}$ and $\{b_n\}$ ($b_n \neq 0$, $n \in \mathbb{N^*}$) with initial term $1$, they satisfy the relation $b_{n+1} = \frac{a_{n+1} b_n}{a_n + 3b_n}$.

(1) Let $c_n = \frac{a_n}{b_n}$, find the general term formula for the sequence $\{c_n\}$;

(2) If the sequence $\{b_n\}$ is a geometric sequence with all positive terms, and $b_4b_6=4b_5b_7$, find the sum of the first $n$ terms $S_n$ of the sequence $\{a_n\}$.

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