
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
(1) Given $\sin \alpha - \sin \beta = -\frac{1}{3}$ and $\cos \alpha - \cos \beta = \frac{1}{2}$, find the value of $\cos (\alpha - \beta)=$ ______. (Write out the calculation process)

(2) Given $\tan (\alpha + \beta) = \frac{2}{5}$ and $\tan \left(\beta - \frac{\pi}{4}\right) = \frac{1}{4}$, find the value of $\tan \left(\alpha + \frac{\pi}{4}\right)$.

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