
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 that a moving circle $C$ is externally tangent to the fixed circle $C_3: x^2 + 2x + y^2 + \frac{3}{4} = 0$, and internally tangent to the fixed circle $C_2: x^2 - 2x + y^2 - \frac{45}{4} = 0$.
(1) Find the equation for the locus of the center $C$ of the moving circle.
(2) If a line $l: y = kx + l$ ($k \neq 0$) intersects the locus of $C$ at two distinct points $M$ and $N$, and the perpendicular bisector of segment $MN$ passes through a fixed point $G\left(\frac{1}{8}, 0\right)$, find the range of values for $k$.

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