
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 an $m$ in the set of positive integers such that $\frac{S_{2m}}{S_{m}} = 28$ and $\frac{a_{2m}}{a_{m}} = \frac{2m+21}{m-2}$, then the common ratio of the sequence ${a_{n}}$ is $\_\_\_\_\_\_.$

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