
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 $f(x)=5\sin^2x-2\sqrt{3}\sin x \cos x+3\cos^2x$.  
$(1)$ Find the interval where $f(x)$ is monotonically decreasing.  
$(2)$ If $f(\alpha+\frac{\pi}{6})=\frac{12}{5}$ and $\frac{\pi}{2}<\alpha<\pi$, find the value of $\tan(2\alpha+\frac{\pi}{4})$.

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