
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 $f(x)=\lg (x+\sqrt{x^{2}+1})$, if for any $x\in (1,2]$, there is $f\left(\frac{x+1}{x-1}\right)+f\left[\frac{m}{{(x-1)}^{2}(x-6)}\right] > 0$, determine the range of the real number $m$.

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