
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 $x$ and $y$ are positive numbers, $\theta \in \left( \frac{\pi}{4}, \frac{\pi}{2} \right)$, and it satisfies $\frac{\sin\theta}{x} = \frac{\cos\theta}{y}$ and $\frac{\cos^2\theta}{x^2} + \frac{\sin^2\theta}{y^2} = \frac{10}{3(x^2+y^2)}$, determine the value of $\frac{x}{y}$.

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