
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
【Definition】In the Cartesian coordinate system, there is a line $x=m$. For any function graph, the points of the graph on the line $x=m$ and the part to the right of the line $x=m$ are moved up by $n$ units, and the part to the left of the line $x=m$ is moved down by $n$ units, resulting in a new function graph. This new function is called the "$n$-shift function" of the original function with respect to the line $x=m$. For example, the "$2$-shift function" of the function $y=x$ with respect to the line $x=0$ is $y=\left\{\begin{array}{l}{x+2}&{(x≥0)}\\{x-2}&{(x＜0)}\end{array}\right.$.<br/>Based on the above information, answer the following questions:<br/>$(1)$ Given that the point $P\left(3,2\right)$ lies on the graph of the "$3$-shift function" of the function $y=kx\left(k\neq 0\right)$ with respect to the line $x=1$, find the value of $k$;<br/>$(2)$ If the quadratic function $y=-x^{2}+2x+4$ has a "$n$-shift function" with respect to the line $x=3$ that intersects the $x$-axis at three points, let $m$ be the sum of the $x$-coordinates of these three points. Is there a positive integer $n$ such that the value of $m$ is an integer? If yes, find the value of $m$; if not, explain the reason.

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