
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 $f(x)=\log_{a}x$ $(a > 1)$, if $b > a$, and $f(b)+\frac{1}{f(b)}=\frac{5}{2}$, $a^{b}=b^{a}$.

$(1)$ Find the values of $a$ and $b$;

$(2)$ Given $k > 0$, if the inequality $kf(x)+\frac{k-1}{f(x)} > 1$ holds for all $x \in (2,4)$, find the range of $k$;

$(3)$ When $x \in [0,1]$, the graph of the function $g(x)=m^{2}x^{2}-2mx+1$ has only one intersection point with the graph of $h(x)=f(x+1)+m$. Find the range of the positive real number $m$.

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