
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 $f(x)=2\sqrt{3}\sin{x}\cos{x}-2\sin^{2}{x}+1$.
$(1)$ Find the value of $f(\frac{5\pi}{4})$;
$(2)$ Determine the minimum positive period and monotonic intervals of $f(x)$;
$(3)$ Compare the values of $f(-\frac{\pi}{5})$ and $f(\frac{7\pi}{8})$, and provide the reasoning.

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