
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 $M$ whose distances to the fixed point $F(\sqrt{3}, 0)$ and to the fixed line $x = \frac{4\sqrt{3}}{3}$ are in the ratio $\frac{\sqrt{3}}{2}$, let the trajectory of the moving point $M$ be curve $C$.
$(1)$ Find the equation of curve $C$.
$(2)$ Let an arbitrary line with non-zero slope pass through point $F$ intersecting curve $C$ at points $A$ and $B$. Does there exist a fixed point $P$ on the x-axis such that the angles $\angle APF$ and $\angle BPF$ are equal?

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