
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 \( f(x) = \sin \left(x + \frac{\pi}{6} \right) + \sin \left(x - \frac{\pi}{6} \right) + \cos x + a \) (where \( a \in \mathbb{R} \) is a constant),

(I) Find the smallest positive period of the function \( f(x) \).

(II) If the maximum value of \( f(x) \) is 1, determine the value of \( a \).

(III) If the maximum value of \( f(x) \) within the interval \( x \in \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right] \) is 1, determine the value of \( a \).

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