
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 \\(\alpha \in (0, \frac{\pi}{3})\\) and \\(\beta \in (\frac{\pi}{6}, \frac{\pi}{2})\\), and \\(\alpha\\), \\(\beta\\) satisfy \\(\begin{cases} 5\sqrt{3}\sin\alpha +5\cos\alpha =8\\ \sqrt{2}\sin\beta + \sqrt{6}\cos\beta =2\end{cases}\\)
\\((1)\\) Find the value of \\(\cos(\alpha+ \frac{\pi}{6})\\).
\\((2)\\) Find the value of \\(\cos(\alpha+\beta)\\).

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