
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 $f(x)=\sin (2x+\frac{\pi }{6})+\frac{1}{2}+a$,

(1) Find the interval(s) where $f(x)$ is monotonically increasing;

(2) If the sum of the maximum and minimum values of $f(x)$ in the interval $-\frac{\pi }{6}\leq x\leq \frac{\pi }{3}$ is $\frac{3}{2}$, find the value of $a$.

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