
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 \( f(x) = \sin \left(x+\frac{\pi}{6}\right) + \sin \left(x-\frac{\pi}{6}\right) + a \cos x + b \) where \( a, b \in \mathbb{R} \) are constants:
1. Find the smallest positive period of the function \( f(x) \).
2. If \( f(x) \) is monotonically increasing on the interval \(\left[ -\frac{\pi}{3}, 0 \right] \) and attains its minimum value of 2, find the values of \( a \) and \( b \).

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