
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 $f(x)=\sin \left( 2\omega x-\dfrac{\pi}{6} \right)+\dfrac{1}{2}$ with a period of $\pi$ and $\omega > 0$,

$\quad\quad(1)$ Find the value of $\omega$;

$\quad\quad(2)$ Find the interval(s) where the function is increasing;

$\quad\quad(3)$ Find the equation of the axis of symmetry;

$\quad\quad(4)$ Find the range of the function $f(x)$ on the interval $\left[0, \dfrac{2\pi}{3}\right]$.

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