
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 $\frac{\cos^2x - \sin x\cos x}{1 + \sin^2x}$;

(2) Find the maximum value and the intervals of monotonic increase for the function $F(x) = f(x) \cdot f(-x) + f^2(x)$.

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