
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 $$f(x) = \sin\left(x + \frac{\pi}{6}\right) + \sin\left(x - \frac{\pi}{6}\right) + \cos x + a$$ has a maximum value of 1,
(1) Find the value of the constant $a$;
(2) Find the set of all $x$ for which $f(x) = 0$.

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