
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 $f(x)=x^{2}-2mx+10$ ($m > 1$).  
$(1)$ If $f(m)=1$, find the expression of the function $f(x)$.  
$(2)$ If $f(x)$ is a decreasing function on the interval $(-\infty,2]$, and for any $x_{1}$, $x_{2}\in[1,m+1]$, $|f(x_{1})-f(x_{2})|\leqslant 9$ always holds, find the range of the real number $m$.  
$(3)$ If $f(x)$ has a zero in the interval $[3,5]$, find the range of the real number $m$.

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