
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
Let's agree: in the Cartesian coordinate system, if a line $l$ passes through the vertex of a parabola $L: y = ax^{2} + bx + c$ (where $a$, $b$, $c$ are constants and $a \neq 0$) and the intersection point of the parabola with the $y$-axis, then we call the line $l$ the "heart-to-heart line" of the parabola $L$. According to this agreement, please complete the following questions:
$(1)$ If the line $y = kx + 1$ is the "heart-to-heart line" of the parabola $y = x^{2} - 2x + 1$, find the value of $k$.
$(2)$ If the "heart-to-heart line" of the parabola $L: y = -x^{2} + bx + c$ passing through the origin is $y = mx + n$ (where $m \neq 0$), is the algebraic expression $\frac{b}{m}$ a constant value? If yes, find the constant value; if not, explain the reason.
$(3)$ When the constant $k$ satisfies $\frac{1}{2} \leq k \leq 2$, find the range of values for the area of the triangle enclosed by the "heart-to-heart line" $l$ of the parabola $L: y = ax^{2} + (3k^{2} - 2k + 1)x + k$ and the $x$-axis and $y$-axis.

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