
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 the function $f(x) = \sin \left(x + \frac{\pi}{6}\right) + \cos x$.

(1) Find the maximum value of the function $f(x)$, and write the set of values of $x$ when $f(x)$ reaches its maximum value;

(2) If $a \in \left(0, \frac{\pi}{2}\right)$ and $f\left(a + \frac{\pi}{6}\right) = \frac{3 \sqrt{3}}{5}$, find the value of $f(2\alpha)$.

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