
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 $f(x)=2\sqrt{3}\sin{x}\cos{x}+2\cos^2{x}-1$ where $x\in\mathbb{R}$.  
$(1)$ Find the intervals where the function $f(x)$ is decreasing, and determine its maximum and minimum values on the interval $[0,\frac{\pi}{2}]$.  
$(2)$ If $f(x_0)=\frac{6}{5}$ and $x_0\in[\frac{\pi}{4},\frac{\pi}{2}]$, find the value of $\cos{2x_{0}}$.

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