
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 H(-3, 0), point P is on the y-axis, point Q is on the positive half of the x-axis, and point M is on line PQ, satisfying $\overrightarrow{HP} \cdot \overrightarrow{PM} = 0$ and $\overrightarrow{PM} = -\frac{3}{2} \overrightarrow{MQ}$.
(1) When point P moves on the y-axis, find the trajectory C of point M;
(2) A line l through point T(-1, 0) intersects the trajectory C at points A and B. If there exists a point E($x_0$, 0) on the x-axis such that $\triangle ABE$ is an equilateral triangle, find the value of $x_0$.

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