
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 a geometric sequence $\left\{a_n\right\}$ where all terms are positive, and $\{a_1, \frac{1}{2}a_3, 2a_2\}$ form an arithmetic sequence, evaluate the expression $\frac{a_8+a_9}{a_6+a_7}$.

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