
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 point P (-4, -4), point Q is a moving point on the ellipse $x^2+my^2=16$ with an eccentricity of $\frac{\sqrt{2}}{2}$ and its foci on the x-axis. Point M is on the line segment PQ and satisfies $\overrightarrow{PM} = \frac{1}{3}\overrightarrow{MQ}$. Find the equation of the trajectory of point M.

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