
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 in the Cartesian coordinate system, the ratio of distances from point M $(x, y)$ to two fixed points O $(0, 0)$ and A $(3, 0)$ is $\frac{1}{2}$.
(1) Find the equation of the trajectory of point M and describe the shape of the trajectory;
(2) Point P $(x, y)$ lies on the trajectory obtained; find the maximum value of $2x^2 + y^2$.

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