
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
If we define: in the plane rectangular coordinate system, the coordinates of point $P$ are $(x, y_{1})$, the coordinates of point $Q$ are $(x, y_{2})$, and the difference between $y_{1}$ and $y_{2}$ forms a new function $y$, that is, $y = y_{1} - y_{2}$. $y$ is called the "number of days function" of $y_{1} - y_{2}$, $P$ is called "day point 1", and $Q$ is called "day point 2". (Dear students: I hope you will not disappoint the time in "counting days" and submit satisfactory answers each time.)  
$(1)$ Given that "day point 1" is point $A(x, kx+4)$, "day point 2" is point $B(x, 2x)$. Point $C(2, 3)$ is on the graph of the "number of days function" $y = y_{1} - y_{2}$, find the analytical expression of $y$;  
$(2)$ Given that "day point 1" is point $M(x, x^{2}+3)$, "day point 2" is point $N(x, 3x)$, $y$ is the "number of days function", find the minimum value of $x+y$;  
$(3)$ For the equation in terms of $x$: ${x}^{2} + (m-2)x + \frac{1}{2}m - 3 = 0$ with two real roots $x_{1}$, $x_{2}$, the "number of days function" $S = S_{1} - S_{2}$. If $S_{1} = 2x_{1}$, $S_{2} = -x_{2}$, and $S = m+1$, find the value of $m$.

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