
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 increasing geometric sequence $\{a_{n}\}$, the sum of its first $n$ terms is $S_{n}$, and $6a_{7}$ is the arithmetic mean of $a_{8}$ and $a_{9}$. Determine the value of $\frac{S_{6}}{S_{3}}$.

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