
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 3rd century BC, the ancient Greek mathematician Apollonius studied many plane trajectory problems in his book "Conic Sections." One of the results he obtained was that the ratio of distances from a point to two fixed points in a plane is a known number, and the trajectory of the moving point is a line or a circle. This type of circle is later known as an Apollonian circle. Given points $A(-2,0)$ and $B(1,0)$ in a plane with a Cartesian coordinate system, and $|PA|=2|PB|$. <br/>$(1)$ Find the equation of the trajectory of point $P$; <br/>$(2)$ If point $P$ moves on the trajectory in $(1)$, and point $M$ is the midpoint of $AP$, find the equation of the trajectory of point $M$; <br/>$(3)$ If point $P(x,y)$ moves on the trajectory in $(1)$, find the range of values for $t=\frac{{y+4}}{{x-6}}$.

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