
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
In a geometric sequence $\left\{a_{n}\right\}$ where all terms are positive, $\left(a_{1}, \frac{1}{2}a_{3},2a_{2}\right)$ forms an arithmetic sequence. Find the value of $\frac{a_{9}+a_{10}}{a_{7}+a_{8}}$.

A) $1+ \sqrt{2}$
B) $1- \sqrt{2}$
C) $3+2 \sqrt{2}$
D) $3-2 \sqrt{2}$

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