
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
Let ${y_1}=\frac{{4{x_1}}}{{x_1^2+1}}$, and $y_{2}=-x_{2}+5-2m$.<br/>$(Ⅰ)$ Find the range of $y_{1}$;<br/>$(Ⅱ)$ If for any $x_{1}\in \left(0,+\infty \right)$, there exists $x_{2}\in \left[-1,1\right]$, such that $y_{2}\geqslant y_{1}$, find the range of the real number $m$;<br/>$(Ⅲ)$ If the inequality about $x_{2}$, $y_{2}(x_{2}-1)\geqslant 1$, has a solution set in ${x_2}∈[{\frac{3}{2},\frac{9}{2}}]$, find the maximum value of the integer $m$.

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