
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 $f(x)=\frac{ax}{x+b}$, and $f(1)=1$, $f(-2)=4$.

$\quad\quad\;①$ Find the values of $a$ and $b$;

$\quad\quad\;②$ Given point $A(1,0)$, let point $P(x,y)$ be any point on the graph of function $y=f(x)\;(x < -1)$, find the minimum value of $|AP|$, and the coordinates of point $P$ at this time;

$\quad\quad\;③$ When $x∈(0,1)$, the inequality $f\left(x\right)≤\frac{2m\cdot\ln x}{\left(x+1\right)}$ always holds, find the range of the real number $m$.

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