
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
A line $l$ with a slope of $k$ intersects the parabola $x^2 = 2y$ at two points $A$ and $B$, and the midpoint of $AB$ is exactly on the line $x = 1$.
(1) Find the value of $k$;
(2) The line $l$ intersects the circle $x^2 + y^2 = 12$ at two points $C$ and $D$. If $|AB| = |CD|$, find the equation of line $l$.

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