
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 $y=x+ \frac {a}{x}$ has the following properties: when $a > 0$, the function is monotonically decreasing in $(0, \sqrt {a}]$ and monotonically increasing in $[\sqrt {a},+\infty)$. Consider the function $f(x)=|t(x+ \frac {4}{x})-5|$ defined on $(0,+\infty)$, where $t > 0$.  
$(1)$ If the function $f(x)$ is monotonically increasing or decreasing respectively in the intervals $(0,2)$ and $(2,+\infty)$, find the range of values for $t$.  
$(2)$ When $t=1$, if the equation $f(x)-k=0$ has four distinct real roots $x_{1}$, $x_{2}$, $x_{3}$, $x_{4}$, find the range of values for $x_{1}+x_{2}+x_{3}+x_{4}$.  
$(3)$ When $t=1$, is there any real numbers $a$, $b$ with $0 < a < b\leqslant 2$, such that the range of values of $f(x)$ in the interval $[a,b]$ is $[ma,mb]$, if there is, find the range of values for the real number $m$; if not, please explain why.

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