
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 ${f_k}(x)={a^x}+k{a^{-x}}$ ($k\in Z$, $a \gt 0$ and $a\neq 1$).  
$(1)$ If ${f_1}({\frac{1}{2}})=3$, find the value of $f_{1}(2)$;  
$(2)$ If $f_{k}(x)$ is an odd function defined on $R$, and $0 \lt a \lt 1$, is there a real number $m$ such that ${f_k}({m{x^2}-mx-1})+{f_k}(m-5)＞0$ holds for all $x\in \left[1,3\right]$? If yes, write down the range of real number $m$; if not, explain the reason.

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