
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 $L$: $y=ax^2+x+c$ ($a$, $b$, $c$ are constants, $abc\neq 0$) and a line $l$ that both pass through a point $P$ on the $y$-axis, and the vertex $Q$ of the parabola $L$ is on the line $l$, we say that the line $l$ and the parabola $L$ have a "one belt, one road" relationship. In this case, the line $l$ is called the "belt line" of the parabola $L$, and the parabola $L$ is called the "road line" of the line $l$.

(1) If the line $y=mx+1$ and the 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 certain "road line" $L$ is on the graph of the inverse proportional function $y=\frac{6}{x}$, and its "belt line" $l$ has the equation $y=2x-4$, find the equation of the "road line" $L$.

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