
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 $f(x)=\sin(2x+\frac{\pi}{6})+\sin(2x-\frac{\pi}{6})+\cos{2x}+a$ (where $a \in \mathbb{R}$ and $a$ is a constant):

(1) Find the smallest positive period of the function and the intervals of monotonic increase.

(2) If the minimum value of $f(x)$ for $x \in \left[0, \frac{\pi}{2}\right]$ is $-2$, determine the value of $a$.

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