
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 $f(x)=\frac{1}{2}\sin 2x\sin φ+\cos^2x\cos φ+\frac{1}{2}\sin (\frac{3π}{2}-φ)(0 < φ < π)$, whose graph passes through the point $(\frac{π}{6},\frac{1}{2})$.

(I) Find the interval(s) where the function $f(x)$ is monotonically decreasing on $[0,π]$;

(II) If ${x}_{0}∈(\frac{π}{2},π)$, $\sin {x}_{0}= \frac{3}{5}$, find the value of $f({x}_{0})$.

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