
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
In the plane rectangular coordinate system $xOy$, for point $P$, point $M$ is defined as follows: if the distance between point $P$ and the origin $O$ is $a$, and the distance between point $M$ and point $P$ is $a$ times $k$ ($k$ is an integer), then point $M$ is called the "$k$ times associated point" of point $P$.  
$(1)$ When $P_{1}(-1.5,0)$:  
① If the $2$ times associated point $M$ of point $P_{1}$ lies on the $x$-axis, then the coordinates of point $M$ are ______;  
② If point $M\left(x,y\right)$ is the $k$ times associated point of point $P_{1}$ and satisfies $x=-1.5$, $-3\leqslant y\leqslant 5$, then the maximum value of the integer $k$ is ______;  
$(2)$ Given right triangle $ABC$ with $\angle ABC=90^{\circ}$, $\angle ACB=30^{\circ}$, $A\left(b,0\right)$, $B\left(b+1,0\right)$. If $P_{2}(-1,0)$, and there exists a point $Q$ on the side of $\triangle ABC$ that is the $2$ times associated point of point $P_{2}$, find the range of values for $b$.

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