
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
(1) Find the intervals of increase for the function $f(x) = \sin(2x + \frac{\pi}{6}) + \frac{1}{2}$.
(2) Given a function $y = a - b\cos\left(x - \frac{\pi}{3}\right)$, where $b > 0$, and its maximum value is $\frac{3}{2}$ and minimum value is $-\frac{1}{2}$ for $0 \le x \le \pi$, find the value of $2a + b$.

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