
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
Define; if the parabola $y=ax^{2}+bx+c\left(a\neq 0\right)$ intersects the $x$-axis at points $A(x_{1}$,$0)$,$B(x_{2}$,$0)$, then we call the line segment $AB$ the chord of the parabola, and the distance between points $A$ and $B$ is called the chord length of the parabola.<br/>$(1)$ Find the chord length of the parabola $y=x^{2}-5x-6$;<br/>$(2)$ Find the range of chord lengths of the parabola $y=x^{2}+\left(n+1\right)x-1\left(1\leqslant n \lt 3\right)$;<br/>$(3)$ Let $m$ and $n$ be positive integers, and $m\neq 1$. The chord length of the parabola $y=x^{2}+\left(4-mt\right)x-4mt$ is $l_{1}$, and the chord length of the parabola $y=-x^{2}+\left(t-n\right)x+nt$ is $l_{2}$. Let $s={l}_{1}^{2}-{l}_{2}^{2}$. Find the functional relationship between $s$ and $t$, and determine the values of $m$ and $n$ such that $s\geqslant 0$ for all values of $t$.

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