
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 \left(0, \frac{\pi}{3}\right)\\), satisfying \\(\sqrt{3}\sin \alpha + \cos \alpha = \frac{\sqrt{6}}{2}\\).

\\((1)\\) Find the value of \\(\cos \left(\alpha + \frac{\pi}{6}\right)\\).

\\((2)\\) Find the value of \\(\cos \left(2\alpha + \frac{7\pi}{12}\right)\\).

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