
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 that the function $f(x)=\sin (π-ωx)\cos ωx+\cos ^{2}ωx\ (ω > 0)$ has a minimum positive period of $π$.
(I) Find the value of $ω$;
(II) The horizontal coordinates of each point on the graph of the function $y=f(x)$ are shortened to half of the original, and the vertical coordinates remain unchanged, resulting in the graph of the function $y=g(x)$. Find the minimum value of the function $y=g(x)$ on the interval $[0, \frac{π}{16}]$.

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