
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) is on the y-axis, and point Q(a, 0) is on the positive half of the x-axis. They satisfy the condition that vector HP is perpendicular to vector PQ. Point M is on line PQ and satisfies the condition that vector PM is equal to vector MQ.

(I) Find the equation of the trajectory C of point M as point P moves on the y-axis.
(II) A line l passes through point T(-1, 0) and intersects trajectory C at points A and B. Point E(t, 0) is on the x-axis and satisfies the condition that the distance between E and A is equal to the distance between E and B. The midpoint of line segment AB is point D, and 4|DE| = $\sqrt{3}$|AB|. Find the value of t.

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