
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 $f(x) = (\sin x + \cos x)^2 + 2\cos^2 x$,  
(1) Determine the intervals where $f(x)$ is monotonically decreasing.  
(2) Find the range of $f(x)$ when $x \in \left[0, \frac{\pi}{2}\right]$.  
(3) Describe the transformation process from the graph of $y = \sqrt{2}\sin x$ to the graph of $y = f(x)$.

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