
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 $f(x)=\sin 2x+2\cos ^{2}x-1$, the horizontal coordinates of each point on the graph of $f(x)$ are shortened to $\frac{1}{2}$ of the original length, while the vertical coordinates remain unchanged. Then, the resulting graph is shifted to the right by $\frac{\pi}{4}$ units, forming the graph of the function $y=g(x)$. Determine the analytical expression for $y=g(x)$.

A) $g(x)=\sqrt{2}\sin x$
B) $g(x)=\sqrt{2}\cos x$
C) $g(x)=\sqrt{2}\sin (4x-\frac{3\pi}{4})$
D) $g(x)=\sqrt{2}\cos 4x$

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