
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
Find the equation of the trajectory of point $P(x, y)$ ($y \geq 0$) in the Cartesian coordinate system $xOy$, given that $P$ is a moving point and its distance to the fixed point $M(0, \frac{1}{2})$ is $\frac{1}{2}$ units greater than its distance to the x-axis.

(1) Determine the equation of the trajectory of point $P$.
(2) If the line $l: y = kx + 1$ intersects the trajectory of point $P$ at points $A$ and $B$, and $|AB| = 2\sqrt{6}$, find the value of $k$.
(3) Suppose the trajectory of point $P$ is curve $C$, and point $Q(1, y_0)$ is on curve $C$. Determine the equation of the tangent to curve $C$ at point $Q$.

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