
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
Determine the values of $x$ and $y$ that satisfy the equation $\frac{x+2y}{x} = \frac{2y}{x+2y}$. Assume $x \neq 0$ and $x+2y \neq 0$.
A) $x = y \pm i\sqrt{3}y$
B) $x = -y \pm i\sqrt{3}y$
C) $x = -y \pm \sqrt{3}y$
D) $x = y \pm \sqrt{3}y$

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