
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
In the Cartesian coordinate system $xy$, points on curve $C_1$ are all outside the circle $C_2: (x-5)^2 + y^2 = 9$, and for any point $M$ on $C_1$, the distance from $M$ to the line $x = -2$ is equal to the minimum distance between point $M$ and points on the circle $C_2$. Find the equation of curve $C_1$.

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