
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 a moving point $M(x,y)$ in the Cartesian coordinate system, let $A(-4,0)$, $B(-1,0)$, and in $\triangle ABM$, $|MA|=2|MB|$.
(I) Find the equation of the trajectory $C$ of point $M$ and the range of values for the perimeter of $\triangle ABM$;
(II) Let $M'$ be the other intersection point of line $MB$ and trajectory $C$. Find the range of values for $\frac{S_{\triangle AMB}}{S_{\triangle AM'B}}$.

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