
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 $f(x)=a\sin\left(2ωx+\frac{π}{6}\right)+\frac{a}{2}+b (x∈R, a>0, ω>0)$ with a minimum positive period of $π$, the maximum value of the function $f(x)$ is $7$, and the minimum value is $3$.

(1) Find the values of $ω$, $a$, and $b$; and specify the monotonically increasing interval of $f(x)$.
(2) Given $g(x)=f(x+\frac{π}{12})$, find the range of $g(x)$ when $x∈[-\frac{π}{3},\frac{π}{3})$.

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