
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
Let $x$ and $y$ be real numbers such that $\frac{\sin x}{\sin y} = 4$ and $\frac{\cos x}{\cos y} = \frac{1}{3}$. Find the value of
\[
\frac{\sin 2x}{\sin 2y} + \frac{\cos 2x}{\cos 2y}.
\]

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