
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 two circles: $x^2+y^2-2x-6y-1=0$ and $x^2+y^2-10x-12y+m=0$.  
(1) What value should $m$ take for the circles to be externally tangent?  
(2) What value should $m$ take for the circles to be internally tangent?  
(3) When $m=45$, find the equation of the line containing the common chord of the two circles and the length of the common chord.

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