
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 $\sin (\frac{\pi }{8}+\frac{\alpha }{2})\cos (\frac{\pi }{8}+\frac{\alpha }{2})=\frac{\sqrt{3}}{4}$, $\alpha \in (\frac{\pi }{4},\frac{\pi }{2})$, and $\cos (\beta -\frac{\pi }{4})=\frac{3}{5}$, $\beta \in (\frac{\pi }{2},\pi )$.

Find the values of:
1. $\cos (\alpha +\frac{\pi }{4})$
2. $\cos (\alpha+\beta)$

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