
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 $f(x)=\frac{1}{2}\cos^{2}x-\frac{1}{2}\sin^{2}x+1-\sqrt{3}\sin x \cos x$.  
$(1)$ Find the period and the interval where $f(x)$ is monotonically decreasing.  
$(2)$ Find the minimum value of $f(x)$ on $[0,\frac{\pi}{2}]$ and the corresponding set of independent variables.

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