
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 $f(x) = \sin 2\omega x + 2\cos^2 \omega x$ where $\omega > 0$ and the smallest positive period of the function is $\pi$.  
$(Ⅰ)$ Find the value of $\omega$;  
$(Ⅱ)$ Stretch the abscissa of all points on the graph of $y = f(x)$ by a factor of $2$ (keeping the ordinate unchanged), then shift the resulting graph to the right by $\frac{π}{3}$ units to obtain the graph of a function $y = g(x)$. Find the interval of monotonic increase of the function $g(x)$.

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