
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
Translate the function $y=2\sin 2x$ by shifting all points on its graph $\frac{\pi}{6}$ units to the right, and then shorten the x-coordinates of all points to $\frac{1}{2}$ of their original length (leaving the y-coordinates unchanged) to obtain the graph of $y=f(x)$. Determine the expression for $f(x)$.

A) $2\sin (x-\frac{\pi}{6})$
B) $2\sin (x-\frac{\pi}{3})$
C) $2\sin (4x-\frac{\pi}{6})$
D) $2\sin (4x-\frac{\pi}{3})$

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