
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 $xy > 0$, vector $\overrightarrow{m} = (2x, 1)$ is perpendicular to vector $\overrightarrow{n} = \left(\frac{1}{2}, -\frac{1}{2}y\right)$, and $x$, $y$, and $2$ form a geometric sequence, find the arithmetic mean of $x$ and $y$.

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