
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 circle with center O, $\overline{AB}$ is a diameter, and $\overline{AC}$ and $\overline{BD}$ are secants such that both intersect at point P. $\overline{AE}$ and $\overline{BF}$ are tangents to the circle at points A and B respectively. If $\overline{AE}=x$ and $\overline{BF}=y$, where $x \not= y$, then the diameter of the circle can be expressed as:
A) $|x-y|$
B) $\frac{1}{2}(x+y)$
C) $\sqrt{xy}$
D) $\frac{xy}{x+y}$
E) $2\sqrt{xy}$

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