
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 the geometric sequence $\{a\_n\}$, $a\_7$ is the arithmetic mean of $a\_8$ and $a\_9$. The common ratio $q$ satisfies the following conditions: In $\triangle OAB$ (where $O$ is the origin), $\overrightarrow{OA}=(1,1)$, $\overrightarrow{OB}=(2,q)$, and $\angle A$ is acute. Find the value of the common ratio $q$.

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