
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 a circle $C: x^2+y^2-6x-8y-5t=0$, and a line $l: x+3y+15=0$.

(1) If the chord intercepted by line $l$ on circle $C$ has a length of $2\sqrt{10}$, find the value of the real number $t$;  
(2) When $t=1$, a moving point $P$ on line $l$ draws two tangents to circle $C$, with the points of tangency being $A$ and $B$, respectively. Is there a fixed point on line $AB$? If it exists, find the coordinates of this fixed point; if not, explain why.

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