
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 circle $M: (x+1)^2 + y^2 = 1$, and circle $N: (x-1)^2 + y^2 = 9$, a moving circle $P$ is externally tangent to circle $M$ and internally tangent to circle $N$. The trajectory of the center of circle $P$ forms curve $C$.

$(I)$ Find the equation of curve $C$.
$(II)$ If the line $y=k(x-1)$ intersects curve $C$ at points $R$ and $S$, is there a point $T$ on the x-axis such that $\angle OTS = \angle OTR$ holds for any value of $k$? If such a point exists, please explain why.

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