
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
Given the function $f(x)=|x-t|+|x+t}$, where $t\in \mathbb{R}$.  
$(1)$ If $t=1$, find the solution set of the inequality $f(x) \leq 8-x^{2}$.  
$(2)$ Given $m+n=4$, if for any $x\in \mathbb{R}$, there exist $m>0$ and $n>0$ such that $f(x)=\frac{4m^{2}+n}{mn}$, find the range of real number $t$.

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