
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 plane $(xOy)$, $F$ is the focus of the parabola $C: y^2 = 2px (p > 0)$, and $M$ is any point on the parabola $C$. When $M$ is in the first quadrant, the distance from the center of the circumcircle of $\Delta OFM$ to the axis of symmetry of the parabola $C$ is $\frac{3}{2}$.

(1) Find the equation of the parabola $C$;

(2) The line $l$ passing through $K(-1, 0)$ intersects the parabola $C$ at two points $A$ and $B$, and $\overrightarrow{KA} = \lambda \overrightarrow{KB} (\lambda \in [2, 3])$. Point $G$ is on the $x$-axis, and $|GA| = |GB|$. Find the range of the $x$-coordinate of point $G$ ($x_0$).

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