
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 the terms of a geometric sequence $\left\{ {{a}_{n}} \right\}$ are all positive numbers, and $\left({{a}_{3}}, \frac{1}{2}{{a}_{5}},{{a}_{4}} \right)$ form an arithmetic sequence, find the common ratio $q$.

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