
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 $f(x)=\frac{1}{2}\sin 2x\sin φ+\cos ^{2}x\cos φ-\frac{1}{2}\sin (\frac{π}{2}+φ) (0 < φ < π)$, its graph passes through the point $(\frac{π}{6}, \frac{1}{2})$.
(I) Find the value of $φ$;
(II) If the horizontal coordinates of each point on the graph of $y=f(x)$ are reduced to half of their original length, while the vertical coordinates remain unchanged, resulting in the graph of $y=g(x)$, find the maximum and minimum values of $g(x)$ on $[0, \frac{π}{4}]$.

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