
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 $f(x)=\frac{[\sin(\frac{\pi}{2}-x)\tan(\pi-x)]^{2}-1}{4\sin(\frac{3\pi}{2}+x)+\cos(\pi-x)+\cos(2\pi-x)}$,
(1) Find $f(-1860^{\circ})$;
(2) If the equation $f^{2}(x)+(1+\frac{1}{2}a)\sin x+2a=0$ has two roots in $x\in[\frac{\pi}{6},\frac{3\pi}{4}]$, find the range of the real number $a$;
(3) Find the maximum value of the function $y=4af^{2}(x)+2\cos x (a\in R)$.

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