
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 $f(x)=2\cos x(\sin x-\cos x)+1$, where $x\in R$.
(1) Find the interval(s) where the function $f(x)$ is strictly increasing.
(2) After shifting the graph of the function $y=f(x)$ to the left by $\frac{\pi}{4}$ units and then stretching the $x$-coordinate of each point on the graph to twice its original length (without changing the $y$-coordinate), the graph of the function $y=g(x)$ is obtained. Find the maximum value of $g(x)$ and the set of $x$ where this maximum value is attained.

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