
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 a moving circle that passes through the point $M(2,0)$ and intersects the $y$-axis with a segment of length $4$, denote the trajectory of the moving circle's center as curve $C$.
(1) Find the equation of curve $C$.
(2) Determine if there exists a specific point $P$ on the $x$-axis such that for any two points $A$ and $B$ on curve $C$, when $\overrightarrow{AM} = λ\overrightarrow{MB} (λ ∈ R)$, the ratio of the areas of $△PAM$ and $△PBM$ is always equal to $\frac{|PA|}{|PB|}$. If such a point exists, find its coordinates; otherwise, provide an explanation.

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