
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 cubic sum formula: $m^3+n^3=(m+n)(m^2-mn+n^2)$.
(1) Find the range of the function $f(x)=\frac{\sin^{3}x+\cos^{3}x}{2-\sin2x}$;
(2) Find the range of the function $g(x)=\frac{\sin^{3}x+\cos^{3}x}{1+\sin2x}$, where $x\in[0,\frac{\pi}{2}]$;
(3) If for any real number $x$, the inequality $\sin^{6}x+\cos^{6}x+a\sin x\cos x\geq0$ always holds, find the range of the real number $a$.

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