
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 $H(-6,0)$, point $P(0,b)$ on the $y$-axis, point $Q(a,0)$ on the positive half of the $x$-axis, and $\overrightarrow{HP} \perp \overrightarrow{PQ}$. Point $M$ is on line $PQ$ and $\overrightarrow{PM} - 2\overrightarrow{MQ} = \overrightarrow{0}$.

(I) Find the equation of the trajectory $C$ of point $M$ as point $P$ moves along the $y$-axis.
(II) A line $l$ passing through point $T(-1,0)$ intersects trajectory $C$ at points $A$ and $B$. The perpendicular bisector of line segment $AB$ intersects the $x$-axis at point $E(x_0, 0)$. If the midpoint of line segment $AB$ is point $D$ and $2|DE| = \sqrt{3}|AB|$, find the value of $x_0$.

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