
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
Reading material: If the roots of the quadratic equation $ax^{2}+bx+c=0\left(a\neq 0\right)$ with respect to $x$ are both integers, then the equation is called a "happy equation." Through calculation, it is found that the discriminant $\Delta =b^{2}-4ac$ of any "happy equation" must be a perfect square. It is defined that $F\left(a,b,c\right)=\frac{4ac-{b}^{2}}{4a}$ is the "happy number" of the "happy equation." For example, the "happy equation" $x^{2}-3x-4=0$ has both roots as integers, and its "happy number" is $F\left(1,-3,-4\right)=\frac{4×1×(-4)-(-3)^{2}}{4×1}=-\frac{25}{4}$. If there is another "happy equation" $px^{2}+qx+r=0\left(p\neq 0\right)$ with a "happy number" $F\left(p,q,r\right)$, and it satisfies $|r\cdot F\left(a,b,c\right)-c\cdot F\left(p,q,r\right)|=0$, then $F\left(a,b,c\right)$ and $F\left(p,q,r\right)$ are called "happy numbers" to each other.  
$(1)$ The "happy number" of the "happy equation" $x^{2}-2x-3=0$ is ______;  
$(2)$ If the quadratic equation with respect to $x$ $x^{2}-\left(2m-1\right)x+m^{2}-2m-3=0(m$ is an integer, and $1 \lt m \lt 6)$ is a "happy equation," find the value of $m$ and find the "happy number" of this equation;  
$(3)$ If the quadratic equations with respect to $x$ $x^{2}-mx+m+1=0$ and $x^{2}-\left(n+2\right)x+2n=0(m$ and $n$ are integers) are both "happy equations," and their "happy numbers" are "happy numbers" to each other, find the value of $n$.

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