
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: In general, the equation $\frac{1}{x}+\frac{1}{y}=1$ is not valid, but some special real numbers can make it valid. For example, when $x=2$ and $y=2$, $\frac{1}{2}+\frac{1}{2}=1$ is valid. We call $(2,2)$ a "magical number pair" that makes $\frac{1}{x}+\frac{1}{y}=1$ valid. Please complete the following questions:  
$(1)$ Among the pairs $(\frac{4}{3},4), (1,1)$, the "magical number pair" that makes $\frac{1}{x}+\frac{1}{y}=1$ valid is ______;  
$(2)$ If $(5-t,5+t)$ is a "magical number pair" that makes $\frac{1}{x}+\frac{1}{y}=1$ valid, find the value of $t$;  
$(3)$ If $(m,n)$ is a "magical number pair" that makes $\frac{1}{x}+\frac{1}{y}=1$ valid, and $a=b+m$, $b=c+n$, find the minimum value of the algebraic expression $\left(a-c\right)^{2}-12\left(a-b\right)\left(b-c\right)$.

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