
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 that $S_{n}$ is the sum of the first $n$ terms of a geometric sequence $\{a_{n}\}$, if there exists $m\in \mathbb{N}^{*}$ such that $\frac{S_{2m}}{S_{m}}=9$ and $\frac{a_{2m}}{a_{m}}=\frac{5m+1}{m-1}$, find the common ratio of the sequence $\{a_{n}\}$.

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