
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
Let circle C be defined by the equation $x^2+y^2-2(t+3)x-2ty+t^2+4t+8=0$ ($t\neq -1$).

(1) As $t$ varies, do the centers of the circles lie on the same line? If they do, write the equation of this line; if not, explain why.

(2) Let line $l: x+y-3=0$ intersect circle C at points A and B. Find the maximum length of chord AB.

(3) As $t$ varies, a series of circles is obtained. Is there a line $m$ that is tangent to all these circles? If such a line exists, find its equation; if not, explain why.

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