
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 that in the rectangular coordinate system, the parametric equations of curve $C$ are $\begin{cases} x=1+\sqrt{2}\cos\alpha \\ y=1+\sqrt{2}\sin\alpha \end{cases}$ where $\alpha$ is the parameter. $O$ is the origin of the coordinate,

(1) Establish a polar coordinate system with $O$ as the pole and the positive direction of the $x$-axis as the polar axis. Find the polar equation of the curve $C$, and determine the relationship between the polar point $A(2,\frac{\pi}{4})$ and the curve $C$.

(2) In the rectangular coordinate system, the horizontal coordinates of all points on the curve $C$ are doubled, and the vertical coordinates are compressed to half of the original. Obtain the equation of the new curve $C'$.

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