
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 a parabola $L: y = ax^2 + bx + c$ ($a, b, c$ are constants, and $abc \neq 0$) and a line $l$, if they both pass through a point $P$ on the $y$-axis, and the vertex of parabola $L$, $Q$, lies on line $l$, then we say line $l$ and parabola $L$ have a "One Belt One Road" relationship. In this scenario, line $l$ is called the "belt line" of parabola $L$, and parabola $L$ is called the "road line" of line $l$.

(1) If line $y = mx + 1$ and parabola $y = x^2 - 2x + n$ have a "One Belt One Road" relationship, find the values of $m$ and $n$.
(2) If the vertex of a "road line" $L$ is on the graph of the inverse proportional function $y = \frac{6}{x}$, and its "belt line" $l$ has an equation of $y = 2x - 4$, find the equation of this "road line" $L$.
(3) When constant $k$ satisfies $\frac{1}{2} \leq k \leq 2$, find the range of the area of the triangle formed by the "belt line" $l$ of parabola $L: y = ax^2 + (3k^2 - 2k + 1)x + k$ and the $x$ and $y$ axes.

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