
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 $f(x)=2\sin({2\omega x-\frac{π}{6}})$ where ${\omega > 0}$ and the smallest positive period of $f(x)$ is $\pi$.  
$(1)$ Find the value of $\omega$ and determine the intervals where $f(x)$ is monotonically increasing.  
$(2)$ Find the maximum value of $f(x)$ on the interval $[{0,\frac{5}{12}\pi}]$.

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