
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 a quadratic function $f\left(x\right)=x^{2}-ax+c$.  
$(1)$ The solution set of the inequality $f\left(x\right) \gt 1$ is $\left(-\infty ,-1\right)\cup \left(3,+\infty \right)$.  
① Find the values of real numbers $a$ and $c$;  
② If the inequality $m^{2}-4m \lt f(2^{x})$ holds for all $x\in R$, find the range of values for $m$.  
$(2)$ If for all $x_{1}$, $x_{2}\in \left[-1,5\right]$, $|f(x_{1})-f(x_{2})|\leqslant 10$, find the range of values for the real number $a$.

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