
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 $f(x) = \sin x + \cos x,$  
(1) If $f(x) = 2f(-x)$, find the value of $$\frac {\cos^{2}x-\sin x \cos x}{1+\sin^{2}x};$$  
(2) Find the range and the interval of monotonic increase for the function $F(x) = f(x)f(-x) + f^2(x)$, where $x \in \left(0, \frac {\pi}{2}\right)$.

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