
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 point $M$ to a fixed point $F(1,0)$ and a fixed line $x=4$, the ratio of the distances is $\frac{1}{2}$. Let the trajectory of the moving point $M$ be curve $C$.

(Ⅰ) Find the equation of curve $C$;

(Ⅱ) Draw any line with a slope not equal to $0$ through point $F$ that intersects curve $C$ at two points $A$ and $B$. Is there a point $P$ on the $x$-axis (not coinciding with point $F$) such that $\angle APF = \angle BPF$? If it exists, find the coordinates of point $P$; if not, explain why.

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