
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 $f(x)=A\sin (ωx+φ), x\in R$ (where $A > 0, ω > 0, 0 < φ < \dfrac{π}{2}$), the distance between two adjacent intersection points of the graph and the $x$-axis is $\dfrac{π}{2}$, and one of the lowest points on the graph is $M(\dfrac{2π}{3},-2)$.

1. Find the analytic expression of $f(x)$ and the interval where it is monotonically increasing.
2. Find the range of $f(x)$ when $x\in [\dfrac{π}{12},\dfrac{π}{2}]$.

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