
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 interval of monotonic increase for $f(x)=\sin(2x+ \frac{\pi}{6})+ \frac{1}{2}$.
2. Given the function $y=a-b\cos(x- \frac{\pi}{3})$ with $b > 0$, if the maximum value is $\frac{3}{2}$ and the minimum value is $-\frac{1}{2}$ within the interval $0\leqslant x\leqslant \pi$, find the value of $2a+b$.

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