
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 the parabola $C_1: y^2 = 2px$ ($p>0$) with focus $F$. There exists a point $G$ on the parabola such that the distance from $G$ to the focus is $3$, and point $G$ is on the circle $C: x^2 + y^2 = 9$.  
(I) Determine the equation of parabola $C_1$.  
(II) It is known that one focus of the ellipse $C_2: \frac{x^2}{m^2} + \frac{y^2}{n^2} = 1$ ($m>n>0$) coincides with the focus of the parabola $C_1$. If there are two different points on ellipse $C_2$ symmetrical with respect to the line $l: y = \frac{1}{4}x + \frac{1}{3}$, find the range of the eccentricity $e$ of the ellipse $C_2$.

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