
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 $\cos\alpha=\frac{3}{5}$ and $\cos\beta=-\frac{1}{2}$, with $\pi < \alpha + \beta < 2\pi$ and $0 < \alpha - \beta < \pi$, find the values of $\cos 2\alpha$ and $\cos 2\beta$.

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