
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
Observe the following group of equations: ①$x^{2}-x=0$; ②$x^{2}-3x+2=0$; ③$x^{2}-5x+6=0$; ④$x^{2}-7x+12=0$; $\ldots$ They have a certain pattern in their roots, which are two consecutive natural numbers. We call this type of quadratic equation "consecutive roots quadratic equation."  
$(1)$ If $x^{2}+kx+56=0$ is also a "consecutive roots quadratic equation," write down the value of $k$ and solve this quadratic equation.  
$(2)$ Please write down the $n$th equation and its roots.

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