
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 $f(x)=A\sin (\omega x+ \varphi )(A > 0,\omega > 0,|\varphi | < \dfrac{\pi }{2})$ whose graph has a highest point at $(\dfrac{\pi }{6},2)$ and a lowest point at $(\dfrac{2\pi }{3},-2)$.
(1) Find the analytic expression of the function $f(x)$;
(2) Find the intervals where the function $f(x)$ is increasing and the equation of its axis of symmetry.

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