
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
Find \(\cos \frac{\alpha - \beta}{2}\), given \(\sin \alpha + \sin \beta = -\frac{27}{65}\), \(\tan \frac{\alpha + \beta}{2} = \frac{7}{9}\), \(\frac{5}{2} \pi < \alpha < 3 \pi\) and \(-\frac{\pi}{2} < \beta < 0\).

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