
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 \( a \) and \( b \) be real numbers such that \( \frac{\sin a}{\sin b} = 4 \) and \( \frac{\cos a}{\cos b} = \frac{1}{3} \). Find the value of
\[
\frac{\sin 2a}{\sin 2b} + \frac{\cos 2a}{\cos 2b}.
\]

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