
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 that the ratio of the distance between the moving point $P$ and the two fixed points $O(0,0)$ and $A(3,0)$ is $2$, and the trajectory of point $P$ is curve $C$.
1. Find the equation of the trajectory of curve $C$.
2. A line passing through the point $(-1,0)$ intersects curve $C$ at points $A$ and $B$. If the coordinates of point $M$ are $(4,0)$, find the maximum area of triangle $ABM$.

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