
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 Cartesian coordinate system, given $A(n,\frac{2}{n}),B(8-m,8-\frac{3}{2}m),C(7-m,0),n,m∈R,n≠0$.<br/>$(1)$ If $m=4$, $P$ is a moving point on the $x$-axis, point $A'\left(2,-4\right)$, when points $A'$, $P$, $B$ are collinear, find the coordinates of point $P$;<br/>$(2)$ If $n=\sin \theta $, $\theta \in \left(0,\pi \right)$, and the angle between $\overrightarrow{CA}$ and $\overrightarrow{CB}$ is $α∈[0,\frac{π}{2})$, find the range of values for $m$.

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