
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 $xOy$, it is known that the distance from any point on curve $C$ to point $M(0, \frac{1}{2})$ is equal to its distance to the line $y = -\frac{1}{2}$.

(Ⅰ) Find the equation of curve $C$;  
(Ⅱ) Let $A_1(x_1, 0)$ and $A_2(x_2, 0)$ be two points on the x-axis ($x_1 + x_2 \neq 0$, $x_1x_2 \neq 0$). Perpendicular lines to the x-axis are drawn from points $A_1$ and $A_2$, intersecting curve $C$ at points $A_1'$ and $A_2'$, respectively. The line $A_1'A_2'$ intersects the x-axis at point $A_3(x_3, 0)$. In this way, $x_1$ and $x_2$ determine $x_3$. Similarly, $x_2$ and $x_3$ determine $x_4$. Given $x_1 = 6$ and $x_2 = 2$, find the value of $x_4$.

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